I don’t see how that is, at least when quantifying over a countable set. A sigma-algebra is closed under countable unions and countable intersections, and that lets us produce events corresponding to quantification over a countable set.
Like, we can talk about the probability that a particular sum of random variables converges, by handling quantifiers in this way.
That seems like handling quantifiers and probability together well.
Now, nesting “the probability of” type expressions, is maybe another story? But I’m not sure that that’s a problem with “extending logic”