Maybe I understand something wrong, but I don't see how we can certainly say anything about the hypotheses.

Each underlying weak-learner has some $A$ that has confidence $1-\delta$ (for some, perhaps, arbitrarily small $\delta$).

If $A$ fails the output might have bad error, for example 0.5. So there might exist a probability that all learning algorithms output bad hypotheses, and the same $h_t$ is chosen.

In general in the analysis of Adaboost we ignored the confidence errors, since we were bounding the error. But here we are talking about Adaboost in a general run…