I am not sure if that really addresses my issue. My problem is that I don't see what eventually forces convergences to 50/50. We can use two or even better many coins or, at least superficially equivalent, one coin and let several experimenters take turns. The results will converge towards 50/50 but only with high probability, or at least I don't see why they alway would, i.e. why not at least one sequence could not converge indefinitely. And that seems to lead to an infinite tower of experiments.

When tossing a coin infinitely often the ratio of heads and tails will converge to 50/50 but only in that sense that you have to toss the coin an infinite number of times an infinite number of times. And you can't really stop here because you may still deviate arbitrarily far from the expected outcome. So you toss the coin an infinite number of times an infinite number of times an infinite number of times. And you have to keep nesting your experiments until you made an infinite tower of experiments.

And I have no idea if that would be sufficient to ensure that you can no longer deviate from the expected outcome, my intuition tends towards no, but then again this is certainly far outside of the realm of things for which I would intuition expect to work. Likely I am really just confused and missing something obvious.

Your problem is that you don't seem to have sufficient understanding of the concept of limits.