First of all, the set you came up with in step 2. is not complete. There's an infinite number of other theories that would explain phenomenon X, which you did not consider. Hence, these theories are not being tested in step 3., and hence you cannot rule them out.
I agree. We can at most, right now, shift the probability of a theory being true by an infinitesimal amount. However, there are two ways of thinking about science. One way is to look at it as a system of confirmation: every time we make a prediction from a theory and it turns out to be correct, we get closer and closer to knowing that the theory is true or, if we keep falsifying theories, we keep getting closer to the correct one.
It is true that this does not seem to be justified with current mathematical frameworks for science. However, not everyone accepts the idea of scientific confirmation, even to any degree. I do not. However, falsification is nice.
Consider a collection of theories {T1, T2,
T2, …, TN} and we use that collection, as a whole, to determine the probability distribution of outcomes for a given experiment. Using little more than probability theory and the assumption tht reality is logically consistent, we can say that if an observation is unlikely, given the probability distribution derived from our collection of theories, then the collection of theories, as a whole, are unlikely to be true.
Of course, how to modify that collection used to make the prediction is tricky. But we continue to make modifications and make predictions until the point where we have run out of reasonable ways to falsify the system.
Secondly, step 3. is in and of itself highly fallacious as well. Just because a theory stands up to some experiment, does not mean it is true.
Indeed, but we can apply parsimony at least. If we have two sets of theories which both explain the data equally well, the most parsimonious one is the most likely to be correct. Once we have shown that two sets of theories explain a data set and have run out of reasonable ways to test them, we just take the simplest collection we have and assume that the system is true, until we are shown otherwise in future experiments (or events in life).
System of Theories
To better explain what I mean by system of theories, consider testing some concept in thermodynamics like the rate of heat transfer. We measure the temperature, using fluid thermometers, in two containers, connected by a conductor over a course of time and plot it to see if our theory holds.
But are we really measuring temperature? No. We are measuring the volume of the fluid in the thermometer, and using theories of material science to infer the temperature. Therefore you are not testing the theory of rate of heat exchange. You are testing the system which includes not only that theory, but the theories used to produce the measurements as well, and if you get a result that's reasonably inconsistent, you are not falsifying a single theory but the collection of theories used to make the prediction.
Robustness of Theory
I would like to add that I generally look at theories as being "robust" or non-robust. A theory is more robust if all reasonable methods to show it false have been made and there are a number of other theories which are dependent on it. The theory body of theory of evolution is important in a number of other fields, such as medical science. If evolution is false, then a lot of our medical theory, including epidemiology, falls apart. It has also gone through a lot of rigorous testing. Therefore it is quite robust. I personally like the word "robust" much better than "true" or "proven" when talking about scientific theories.