> The nuance here is that we have to somehow be able to distinguish 0.5 - eps from 0.5 for very small epsilon.

If you ignore the problem then the problem indeed goes away. The need for distinguishing very small epsilon exists because of the continuity assumption, and because of the continuity assumption you can't really solve it either.

> Then we can decide to go left if it tells us x < 0.5, and right if it tells us unsure or that x >= 0.5.

Now you just moved the problem to deciding at which point you are unsure. As long as there is a decision to take the issue persists, it's only if you always go left (or always go right) that the issue doesn't exist.