When sampling a population for surveys we can often limit our sample size to hundreds, but when doing a Monte Carlo simulation we need way more. Why? I’m a bit of a stats-noob, so I am not sure I will manage to formulate this question properly, but let me do my best.
I‘m trying to develop an intuition for sample sizes and when they are sufficient to get to a reasonable degree of representativeness of a larger population.
I know that when doing surveys or polls, a sample size of mere hundreds or thousands is often sufficient, even for very large populations. (This is at least what calculators such as this one are telling me.)
So now what happens if we extend this notion to Monte Carlo simulations of a multi-parameter model? Every simulation can be thought of as representing a sample of the larger population of all possible simulations (the solution space).
The solution space of a multi-dimensional problem can be very large, but the survey/poll scenario tells me that this doesn’t mean that the sample size will need to be large.
This seems to suggest that any application of the Monte Carlo method can be concluded within several hundred/thousand of simulations as well. But I feel this cannot be true, as it would appear to defeat the purpose of the plethora of “more efficient alternatives” to Monte Carlo (such as Markov Chain Monte Carlo).
What am I missing? What is the difference between surveying a large population of people on their election vote (for example) and “surveying” a large population of parameter permutations on their performance (when plugged into a certain model)?
Why does a surprisingly small number of surveys suffice to get a decent idea of the likely outcome of an election, while a small number of Monte Carlo simulations does not suffice (I think) to get a decent idea of the likely (top) performance of a model?
My apologies for needing too many words, but it’s the best I can do I am afraid. I hope you catch my drift regardless and can steer me in the right direction. Thanks a lot!

One important side note:

This seems to suggest that any application of the Monte Carlo method can be concluded within several hundred/thousand of simulations as well. But I feel this cannot be true, as it would appear to defeat the purpose of the plethora of “more efficient alternatives” to Monte Carlo (such as Markov Chain Monte Carlo).

Markov Chain Monte Carlo is almost always not more efficient than a Monte Carlo approach with independent samples, but rather much, much less efficient. With most MCMC algorithms, each sample is positively correlated with the previous sample. The impact of this correlation can be quite extreme: it's not too uncommon to see every one thousand samples being equal to one new sample (i.e. takes one thousand samples to forget the previous state in the MCMC algorithm).
In these scenarios, if we you were content with the inference learned from 100 independent samples, you would need 100,000 samples from an MCMC algorithm with the heavy correlation mentioned above.
MCMC is generally used when you cannot take independent draws from the target distribution.
