The long answer is that most mathematicians would consider them identical in the sense they endorse the notion of independent events. A sequence of coin tosses of the same coin n times is in terms of probability is the same as n coins tossed if one takes the a sequence of physical locations to build a sequence out of the second case. What you are digging at is why.
So, a philosophical analysis would revolve around your vocabulary use. To you, it's "problematic" and "unproblematic" when one is dealing with a sequence of heads versus when one is dealing with a collection of coins tossed simultaneously arranged in a sequence. To wit:
In the first experiment, throwing the same coin N times and getting an outcome of all heads is increasingly problematic for the same fair coin... Whereas for the second experiment, getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole outcome.
Your first statement is the gambler's fallacy and the second statement is the fallacy of composition. So, let me say a few words first, to reorient the conversation.
First, "problematic" has no definition. As a debater, I would mock for you using this term in mathematical discourse to get laughs. But here, my goal is not to win, but to persuade you that "problematic" does not exist in a math dictionary in any sense connected to frequentist notions of independence. (Of course, feel free to produce a counterexample.) See, were this a debate, the argument is already 80% over, because your first two statements are well-known fallacies. But, since this isn't a debate, let's explicate on what is happening with your thinking.
Your term "problematic" is a red flag. Problematic is a very ambiguous, lay term that expresses normativity without a framework. If mathematicians used the term "problematic", there would be a rigorous definition argued over the community and available. Do a search for the term "problematic" in the WP article on independent events. Know how many hits? Zero, zilch, zip, none, bupkis. That's a clue that you're dealing with idiosyncratic idiolect or vagueness or novelty leading up to a neologism. If problematic has no generally accepted mathematical usage as jargon, what should be made of it?
I would simply argue you are using your intuition, and doing so reasonably. Some philosophers hate intuition. That want deductive certainty. But deductive certainty or worse, certainty by faith applied to real-life problems is a poor choice. That's not to say intuitions are always right. Certainly not. I once went to a riverboat with a pet "theory" based on an intuition, that doubling a bet would be a good strategy at roulette. My reason was the same as yours. It would be increasingly unlikely for the next flip to come out red after a long sequences of black. Mein was a Gedankspiel, so I simply wagered in my head for an hour. I was overwhelmingly in the negative based on my intuitions of probabilities of sequences. And that's not because I'm stupid, but because my brain, like most brains faces cognitive biases. Simply put, there is a psychology to misrepresentations like fallacy. The WP article on the gambler's fallacy has an entire section devoted to it.
Cognitive Biases, Fallacies, and Statistics
Yes, our brains are hardwired to think just as you claim, that increasingly long sequences of the same or similar values influence continued results. But it's an illusion, and here's why. on the one hand, you can view a sequence of n tosses as 1/2^n which are very special. They're all heads! Whereas the remainder of non-all heads sequences is 2^n-1 and are not special patterns. So it seems odd to have 1/2^n then 1/2^(n+1) and then 1/2^(n+2) all happen in a row. Doesn't feel right to have three rare events happen in a row, does it. But there's another interpretation.
The laws of physics simply don't care (on a fair coin) what is a head or tail. There are only two sides, and flip and come what may. You see, physics is blind to whether it is heads or tails. A fair coin flipped in sequence is just randomly determined by stochastic processes, and by definition they cannot be affected by the orientation of the coin upon landing. The final orientation is incidental the degrees of freedom which govern the motion. There's the hand flip, the wind, the Coriolis effect, resistance to the wind, variations of the distance from the center of Earth's gravity, etc. But BY DEFINITION, the orientation of the coin has no role in shaping the motion because the orientation of a coin isn't a physical property, it is a mental one of the observer.
Let's try another example. Fill a plastic bottle with 50 red marbles and then with 50 green ones on top. Two layers. Shake. And repeat. And repeat. And as long as you are alive, the odds of you shaking them back into two layers are so theoretically minuscule, that with confidence one can say it won't happen. This is the nature of entropy. Why? Because the color of the marble, isn't a physical property, strictly speaking. Colors are constructed by our bodies and brains, at least according to theories of embodied cognition. To a red-green blind person, there isn't even an experiment here to be had. And the universe is no person at all. In the same way, when you flip a fair coin, the randomness that inheres in the universe conducts that coin into a set of frequencies that can be demonstrated empirically over and over.
When I attended a university once long ago, our teacher had us in pairs create a a really long sequence of random numbers with our intuitions, and then with a coin. The cognitive bias towards our intuitions is so strong, he plugged the sequences into software and could tell for almost every group in the class, which of ours was done with a coin, and which we made up. It seems a fantstic claim, but it's actually one of the central theses of Thinking, Fast and Slow. Our brains are just really bad at statistics, and as Kahneman points out, his research demonstrates that extends to professional statisticians! (The man won a fancy prize, and the book is worth a read.)
A Closer Look at the Second Fallacy
getting an outcome of all heads, is not problematic for any coin, simply thrown once, thus not problematic for the whole...
The fallacy of composition is an informal fallacy that arises when one infers that something is true of the whole from the fact that it is true of some part of the whole.
You almost nailed it word for word. ;) What's the argument here? In fact, there's an easy counterargument from math. Measures and measures of central tendency are not the same thing. But yet, we again are capable of biases that mislead us. If I tell you that a set of integers has a mean, median, and mode of 50, and that the values go from 1 to 100, what can we infer about the underlying set? Not much. But our minds will try to fill that set. We can have ten values of 50. We can have a set with an outlier, and a skew on the other side to offset it. We can a sequence leading to 49, 2 50's and a sequence starting at 51. We simply don't know, and our brains our evolved to know. So the brain takes a guess, or fills in the blanks, or sees a picture of your favorite deity in your breakfast toast.
The gambler's fallacy is a well established empirical fact. And event independence is also a well established mathematical and empirical fact. And our logic, when we become specious, is often lead astray by our intuitions. Casinos make billions, and the lotteries are called taxes on the mathematically illiterate because the findings of statistics are pretty dang reliable. What you need to ask yourself is why do you think you can out argue hundreds of years of peer-reviewed, empirically validated statistical research? Why do you believe that intuitions that arise from your cognitive dissonance somehow invalidate the categories of fallacy created and accepted by logicians all over the world? When you are confronted with the fact that your words contradict (a sequence of fair coin toss by definition is random, and your knowledge of it doesn't impact that sequence), why do you scamper to find better words to affirm your feeling of distrust of the claim? That's easy! It's human nature, and we all do. Some of us just learn to spot the misrepresentations like the gambler's fallacy, the fallacy of composition, hasty generalization, and so on. Trust me, if longer and longer sequences were "problematic", I would have cleaned out the casino at the roulette table long ago. ;)