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!

I have doubts that the statement "a surprisingly small number of surveys suffice to get a decent idea of the likely outcome of an election" actually holds well in practice. Well, maybe depends on your definition of "decent"...
Caveat: What I know about polling is mostly from reading FiveThirtyEight. If you want to learn more about the accuracy of polls, in the US at least, take a look at some FiveThirtyEight posts tagged "polling accuracy".
To start with, you are considering a very simple problem: estimate a proportion by sampling a homogeneous population. Voting intent on the other hand depends on a number of factors (personal characteristics, the state of the economy, other current questions/concerns relevant to voters). So polling companies don't survey respondents (completely) at random: they want to have samples that are representative of the population (in a county, a state or a country). So surveys are designed by using census data to determine how to construct a representative sample efficiently. The sample itself can be biased (it may not be efficient to over-sample the most common sub-group of voters) but the bias can be corrected to calculate an unbiased estimate of voting preferences.
Another point to consider is that when an election is close, it's probably necessary to have a smaller margin of error and therefore collect a larger sample of respondents.
The second part of your question about Markov chain Monte Carlo methods has a similar issue of making a simplistic generalization. No one wants to waste their time and computational resources on running their MCMC sampling for longer than necessary to get convergence. That's why it's so important to have tools to diagnose convergence issues. Some chains may need 1,000 samples; others 1,000,000.
