The central limit theorem states that if we take a take a large enough sum of random variables, the sum will approach a normal distribution. I am confused about why we focus only on the sample size and not the number of samples. The way I'm thinking about it, when we take a sample of random variables then calculate the sum, that sum is just a singular number (we "compress" the sample into one statistic). If this is correct, then don't we need more samples to get more sums so we can actually plot the distribution? If we only calculate one sum (say with a sample size of 100,000 random variables), then we only have one number which is clearly not enough to create a distribution.
the CLT is just telling you how the average of the sample is distributed. Just as if I tell you the throws of a die are distributed equally with probability 1/6, or that a particular coin is fair (50% heads, 50% tails)
how you check this is a different issue eg by throwing the die multiple times and creating a histogram.
Actually, in practice, when we conduct a survey/experiment. It is costly and takes time to reconstruct the same thing with the same controlled factors many times. Therefore, conducting one time with a large enough sample size will produce the desired results, where we can test the hypothesis under the CLT. Furthermore, some hypotheses based on observed data (eg economic and social data) cannot be reproduced. So, the sample size will give us more advantages than the number of samples.