Simple formulas only work because the models themselves for those polls are incredibly simple and adding a bit more complexity requires a lot of tools to compute these uncertainties (this is part of the reason you see probabilistic programming so popular for people doing non-trivial polling work).

There are no simple approximations for a range of even slightly complex models. Even some nice computational tricks like the Laplace approximation don't work on models with high numbers of parameters (since you need to compute the diagonal of the Hessian).

A good overview of the situation is covered in Efron & Hastie's "Computer Age Statistical Inference".

If you don't have an analytical expression for your asymptotic variance, you do have to use bootstrap though.

For public opinion polls, the estimator is simple (i.e., a sample mean), so we have an analytical expression for its asymptotic variance.