>Forecasts give us probability distributions based on historical polling error data.

They would give you a probability distribution if they were truly random samples. But in reality, response rates are very low, and pollsters make up for this using demographic weighting and modeling - so the results don't actually represent any real population, but rather are extrapolated from people who respond onto a hypothetical population that the pollster thinks is likely to vote. This is why they are spectacularly and systematically off, more and more over time. It's not just variance.