This is the part most intuitively ignore and therefore fail:
-Probability of finding "a": 1/16
-Probability of finding "ab": 1/256
-Probability of finding two "ab" in 256 attempts: approximately 0.18
As you can see, if you find "ab" in an early shot, the probability of finding another "ab" in the next 256 attempts is very low. This would allow you to skip those subsequent attempts without losing significant precision, since although each attempt is still an independent event, the chances of it being there are very low.
Here's your fallacy (which goes in repeat mode with you):
You didn't take into account that the probability to not find "ab" at all in 256 attempts is 36%.
You didn't take into account that the probability to find "ab" more than once is not "very low", but at a good 28% (100 - 36 (for 0) - 36 (for 1)). Because you never added the probabilities for it to appear 3 times, 4 times, 5 times, and so on (up to 256 times). You only base some claims on the fact that it's very unlikely to appear "a second time", but
it can also appear more than 2 times, not necessarily
just one more time.
So while in principle you may be on to something (though nothing more than simply
observing the normal behavior of a uniform random variable), your calculations are off quite a bit, to the point that the combined probabilities of
failure once you go skipping over and over again, accumulate, and they accumulate in the fashion that you like really much: compounded.