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 538.com 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.

Another issue 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 chains 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.

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.