What actually breaks in probability when you let $\mathcal F = 2^\Omega$ for uncountable $\Omega$? 

To the best of my knowledge, this is not a duplicate question. While there are similar-sounding questions, answers to them usually cite:

1. Vitali sets. However, only translation-invariant measures prevent Vitali sets. Probability measures don’t need to be translation-invariant, so Vitali sets are irrelevant. You can totally have a probability measure and $\sigma$-algebra containing Vitali sets.
2. Lebesgue measures. Same objection as above.
3. Banach-Tarski. But measures must be at least rotation-invariant to prevent Banach-Tarski (or live in 2-D). Probability measures don’t need to be rotation-invariant, so Banach-Tarski is irrelevant.

AFAICT, there seems to be a logical gap in the usual explanations. 

Could someone provide a concrete example of any paradoxes that emerge from uncountable sets that apply to probability theory? Any issues with letting $\mathcal F = 2^\Omega$? 

Perhaps there is no such paradox. For example, $\delta\_a(\cdot)$ seems like a perfectly good probability measure that works with $\mathcal F = 2^\Omega$. The answer might “There’s no paradox, but here is an extra property X we want from our probability measures, here’s why it matters, and for this restricted class of measures, they cannot be defined on $\mathcal F = 2^\Omega$.” If so, I’d love to know what this property $X$ would be.

Apologies if I’ve made an error in this post; I’m new to measure theory!