So I’m as much of a lay person on this as you. Still, it would seem to me just on the face of it that there are more questions to consider here than just the size of functional sample space as a fraction of the size of the total sample space.
Suppose you were to toss a coin with two distinguishable circular faces with radius r, and a cylindrical mantle of height h. In principle, there is nothing to prevent it from landing on the side that wouldn’t also prevent it from landing on either face. Between the mantle and the two faces, there are three possible states it can land in, and one state we are concerned with. Does it therefore follow that the probability of it landing on a face is 2/3, or that the probability of it landing on the mantle is 1/3? Well, we could construct a particularly tall coin, more of a cylinder, whose probability to land on the mantle actually is 1/3. But if we are comparing the raw number of states to arrive at the probabilities, then we should predict it to be 1/3 irrespective of r and h. To fully characterize a probability space, therefore, we need not only the number of states in the sample space, but also the probability measure, a function that assigns weights to all subsets of the sample space.
But let’s grant you this, and say that all amino acid sequences are equally likely to occur prima facie. In the end we are concerned with the probability to move from any functional sequence to another functional sequence (already a smaller space of starting points, mind you), and how this compares with the probability to move from any functional sequence to any non-functional sequence. We could again only consider the number of states in each subset of the sample space. This would be fair consideration assuming that gene sequences are swapped out completely and arbitrarily between one generation and the next. However, if this were so, then genes would not correlate between parent and offspring, and could therefore not be the carrier of heritable traits.
From what I understand, what happens instead most times between generations are individual mutations of but a handful of base pairs at a time. So while the number of different amino acid sequences one amino acid sequence is free to mutate into is as large as you say, in order to say how far the new sequence may be from its parent’s we need to know the probability p that a mutation occurs between the two generations. The probability that multiple mutations occur in the same generation could be a lot lower than p still, depending on how correlated or anti-correlated mutations are.
Then we need to consider that not every single-point mutation actually changes the amino acid the given codon produces, since many amino acids correspond to more than one codon each. Then we need to consider that no amino is as close to all the others as a single base pair, anyway. So even if we are swapping to a different amino acid with our mutation, we are not swapping to just any of the 19 others, but rather to one of a select few that are accessible in a single step from the one we started with.
In the end, the probability to access completely foreign amino acid sequences should be rather very, very low, and the probability to access neighboring ones should be much higher. And since the one we started with already produced a viable organism, it may not be out of the question that the very similar amino acid sequences it can mutate into for the next generation may very well still also produce viable organisms. A few might not, but if the next organism is not viable, then it will be a dead end whose genome we are unlikely to recover many generations after the fact. Many others may produce viable organisms but with disadvantagious traits for the current environment. Again, those lineages are unlikely to survive in the long run, unless the environment they find themselves in changes to support their variant before extinction.