It depends. If you're hacking on free software, as you should be, then after the first iteration, everyone can start using the clearly-much-better thing that is already working, rather than wasting months finishing their own inferior versions. And the person who wins the race that time around may not be the same one who won the first time.

The other thing is that there are distributions like the Cauchy distribution that are so heavy-tailed that they don't even have a mean†, or even a variance. The Law of Large Numbers doesn't apply to them at all! And even for more ordinary heavy-tailed distributions that do have means, like the lognormal distribution (relevant here since it's empirically the distribution of how much we misestimate tasks by) the Law of Large Numbers takes a lot more than "a few times" to start making the distribution of the sum look normal. Try it! Pop open Jupyter and convolve the lognormal distribution with itself a few times! How long does it take before it even starts to look Gaussian? How long does it take before it still looks Gaussian in a log-linear plot?

†In your comment you said "mean and expectation", but those are the same thing. Possibly you don't know anything about statistics yet? If so, welcome! You're one of today's lucky ten thousand!