10,000 divided by 10 is 1,000, which isn't exactly a big sample size for each country. If you wanted to get a general idea of the political makeup of the region, that's probably fine, but if you wanted to really drill deeper, seems like a bigger sample size would be preferred.
However, the point is that this argument has no actual mathematical basis. From a maths point of view, there's no provable benefit in having a larger sample size, as long as your selection is truly random. In statistics "total population size" isn't even a variable in your equation to work out the needed sample size. The only reason for big sample sizes in political studies is just that - political - because big numbers are more convincing to people who never actually studied statistics 101.
e.g. imagine a bag with X number of black marbles and Y number of white marbles. You pull out marbles one by one, note their color and estimate what percentage of the bag is black marbles or white marbles. The total number of marbles in the bag is
entirely irrelevant to how many marbles must be drawn to determine the proportion of black vs white to a specified level of accuracy. e.g. if I double the number of marbles in the bag, the accuracy of the estimate based on 100 marbles is still
exactly the same as it was before. The bag could hold a billion or a googleplex worth of marbles, and it wouldn't in fact change your accuracy after drawing 100 marbles even one bit. And that's because total population size
isn't a variable that affects
proportions, which is all that the statistic is measuring.
The only correct argument here is - is the sample selection truly random? If your sample selection mechanism
isn't random then it won't actually make
any difference if you sample 1000, 10000, 100000 or 1000000. It will still have the exact same bias. e.g. if everyone I sample comes from Paris and I'm claiming that this is representative of all Europeans, increasing the sample size won't fucking improve the metric even one bit. Ok, I'll ask
twice as many Parisians what they think. Now is it more "representative" of Europeans? Or, you could ask 250 million people, half of Europe, but if you had a biased selection (e.g. focused on big city people) then it's going to be worse than a carefully selected sample of 1000.