This is an interesting question, and there have already been some good answers (especially those by @Dcleve and @bumble), so I thought I would just give a brief review/Fisking of the video that is linked at the end of the question (which is well worth watching).
The video is by Sabine Hossenfelder†, who is a working theoretical physicist (Google scholar profile) and science communicator. Her YouTube channel is well worth subscribing to - it is very good and I'd also recommend reading her book "Lost in Math - How beauty leads physics astray". However, when watching the video there is an important piece of background information, which is that Hossenfelder is a fundamentalist Popperian who appears to believe that falsification is the absolute demarcation between science and non-science, which is not something I would agree on. I would view science as a search for the best naturalistic explanation for reality (including the bits we can't observe). I would argue that a falsifiable naturalistic explanation was a better explanation than one that wasn't, but I don't see it as a reliable means demarcation. I see research on multiverses as being obviously scientific, Hossenfelder does not. However, just because an argument is not a scientific one, doesn't mean it is wrong.
"According to this idea [fine tuning] it is extremely unlikely our
universe would just happen to be the way it is by chance, and the fact
that we nevertheless exist requires an explanation"
I would disagree that this requires an explanation. It is possible that there is no reason that the universe is the way it is. However an explanation for the universe that also explains why the constants have the values that they have is arguably a better explanation than one that doesn't... unless, of course, you happen to be a fundamentalist Popperian and require the explanation of the constant values to be falsifiable.
While Hossenfelder mentions religion, I am going to steer clear of that and stick to multiverses as the example as religion is likely to trigger cognitive biases that just get in the way of discussion of the validity of the argument.
"the same argument is used by physicists to pass of unscientific ideas
like the multiverse or naturalness as science"
Note "pass off" is a rather loaded wording, and the "unscientific ideas" is predicated on a strict Popperian view of what science is, which is not necessarily correct.
"There is a long list of calculations of this type [that allow
constants that do or don't support life], but they are not the
relevant part of the argument"
I think this is actually quite an important point. We have theory/calculations to support the plausibility of other sets of values of constants, but no evidence, so this is part of our prior beliefs. We can state our premises, but whether the conclusion is correct depends on whether the premises (theory/calculations) are correct. Any argument constructed from priors, can only really tell you whether is is reasonable to believe the conclusion if you accept the premises (but then again that is no different to predicate logic).
"particle physicists use the same argument when the ask for the next
particle collider"
To some extent that is true, but it is also implying that particle physicists are using an argument for financial reasons rather than scientific ones. This seems a bit of an ad-hominem attack that doesn't really work for me.
"they claim it requires explanation why the mass of the Higgs boson
happens to be what it is"
So again, this seems to boil down to what you think science is about, if you think that science is a search for the best explanation, then of course it requires explanation, because everything requires better explanation, especially if we don't have an explanation at all.
"what is wrong is the claim that the constants of nature that we
observe are unlikely. There is no way to quantify this probability
because we will never measure a constant of nature that has a value
other than the one it does have. If you want to quantify a
probability you have to collect a quantity of data."
This is incorrect. You do need data to empirically estimate a probability, but that is not the only means of quantifying a probability. Another way to quantify a probability is to calculate it using a model.
"You could do that, for example, if you were throwing dice. Throw
them often enough and you get an empirically supported probability
distribution."
The problem here is insertion of "empirically supported" as if you couldn't have a probability distribution without empirical support. But this isn't the case, most often when people say "this is unlikely to happen by random chance", they are talking about a probability quantified by a model, not by an empirically determined distribution. This seems to me to be an expression of Hossenfelder's rather extreme position on what science is - it is all about empirical evidence.
"but we do not have an empirically supported distribution for the
constants of nature"
yes, but we do have theoretical/model based distributions on which the argument could be based.
"And why is that, ... , because they are constant. Saying that the
only value we have ever observer is unlikely is a scientifically
meaningless statement"
This is again dependent on whether you share Hossenfelder's extreme falsificationist view of what science is. Others have no problem with the use of models in science to determine likelihood, it is what is usually meant by "caused by random chance" (not that random chance is an explanation for anything).
"we have no data and will never have data which allow us to quantify
the probability of something we cannot observe. There is nothing
quantifiably unlikely therefore there's nothing in need of explanation"
Again, this is entirely contingent on Hossenfelders position on what science is (search for explanation - verses falsifiability) and whether probabilities can only be quantified by empirical estimation rather than by statistical models.
"If you look at the published literature on the supposed fine-tuning
of the constants of nature, the mistake is always the same. They just
postulate a particular probability distribution". It is this
postulate that leads to their conclusion"
This is not a mistake, depending on what you are trying to argue. If you are arguing that it is reasonable to believe the theory on which the distribution is postulated, then it is reasonable to conclude that it is likely we live in a multiverse. If you state it as an unconditional conclusion about the nature of reality directly, then it is a mistake.
I think most usages of "fine tuning" arguments are the former, Hossenfelder seems to treat them as the latter, perhaps because that is more in accord with her view of what science is about?
"logical fallacies... begging the question"
Absolutely, if this was interpreted as evidence of what is real, rather than what we should believe. Note the theory used to postulate the distribution will have some evidence in the form of consilience - what other observations we can make that do not have a good explanation in the absence of that theory.
"they pick a particular distribution that makes it unlikely"
That seems a little unfair. The particular distribution is based on a theory that is explicitly stated and open to criticism. Suggesting they chose their theory to fit in this argument is an insinuation, they formulated a theory that explains other evidence as well.
"they could as well pick a distribution that make the observed values
likely"
but would it explain anything else? Consilience is part of what makes an explanation a good explanation. Who cares if you can pick a distribution that makes the observed values likely if it depends on a theory that can't explain any other theory?
"not all arguments are unscientific, the best known is [balancing a
pen on its tip] if you saw that you'd be surprised ... you'd look for
an explanation, a hidden mechanism ... but the balanced pen is a
different situation. The claim that the balanced pen is unlikely is
based on data"
Yes, we would intuitively see this as unusual as it doesn't fit with our experience (data) of how physical objects behave in the Earth's gravitational field.
However, given Newton's laws etc. we could have determined from first principles that a pen is unlikely to be in a balanced position in gravitational field without ever having observed any similar situations. We could work out for ourselves that fine-tuning was necessary. This is very much like having a theory of how constants are determined and using that as a basis for a probability distribution.
Again, this seems more a statement about Hossenfelders heavily empirical position on what science is.
Ironically ISTR pen balancing was the example Alan Guth used in his "popular science" book on his work on inflationary cosmology. It is very unlikely that we would see a universe that was so close to exactly flat as the one we observe, given what theory we had on the subject, so it "required" an explanation. Inflation gives a plausible explanation for why it is flat, but it also explains quite a lot of other things as well. This is how science works if you view it as a search for the best naturalistic explanation. The more a theory explains with reasonable plausibility, the better the explanation (all things being otherwise equal).
"You can alternatively [rather than frequentist] intepret the term
"unlikely" using the Bayesian interpretation of probability. In the
Bayesian sense, saying that something you observe is unlikely means
you didn't expect it to happen. But with the Bayesian interpretation,
the argument is that the universe was especially made for us doesn't
work. That is because in that case it is easy enough to find reasons
for why your probability assessment was wrong and nothing is in need
of explanation"
This is missing the point. Under the Bayesian interpretation, those arguing for the multiverse have set out exactly what their arguments and assumptions are. If you want to disagree, then you provide a better theory that explains all of the observations that their theory explains (not just the fine-tuning) and use that instead. If you cherry pick an counter-theory simply to refute the fine-tuning conclusion, then you are the one begging the question.
If you view science as a search for the best naturalistic explanation, then it is about comparing and relative evaluation of explanations. If you are a hard-line falsificationist it is all about whether something can be falsified and trying to falsify. Hard-line falisficationism is more strongly skeptical of everything, but I think it can be taken too far. For example
"[the multiverse is an assumption that is an unnecessary addition to our theory of the universe]"
Again, whether it is necessary depends on what you think science is about. I'd say it was more a hypothesis that is suggested by our existing theory of the universe and worthy of consideration.
"but this does not mean the ... multiverse does not exist, it just
means evidence cannot tell us whether they do or do not exist [which
means it is not a scientific idea]"
Completely agree with the first bit. However, all of our knowledge will be a mixture of experience and theory (Kant?), this is a spectrum from purely experience (we exist to observe the constants) to completely theory (multiverse). The last bit is again more an assertion of what science is, which is not universally agreed to be hard-line Popperianism.
So there is some value in Hossenfelder's video, but there is also a fair amount that can be reasonably questioned. I find her YouTube channel very thought provoking, which I think is what it is for, even if you don't agree with what she presents.
† I am not sure what Hossenfelder's title is, so I will just refer to her as "Hossenfelder" rather than "Prof. Hossenfelder" or "Dr. Hossenfelder", but this is not done out of disrespect for her qualifications or experience.
