The Central Limit Theorem needs the prerequisite that your distribution has a finite variance. It's not at all clear that this holds for the return of startup investments. In more practical terms, when the variance is very high or there are outsized impacts from small portions of your population, it can take arbitrarily many samples before your average starts to converge. So it's entirely possible that sampling 100 companies isn't enough.
Statistically the central limit theorem is assuming sampling from the same population, there is no guarantee that two startup indexes are actually sampling from the same population
Lots of conditions that may not be satisfied. Independence is a very hard one. results may be well correlated even if they look like they should be independent