you don't understand random sampling, right ?
Firstly you have to understand what you are speaking about when talking about sampling. Asking for 200 "fixed number" sampling for most analyzes is NEVER done. Why is that ? because for most analyzes we can estimate the probability that the population considered. Why is that ? Because of the work of previous scientist those guy you dismissed.
That is why we know we do not need to take 200 sample of a cylinder to check the metal mass spectro. That is why we do not extract 200 blood probe from your blood. And that is why you do not take 200 radio carbon sample from the cloth. Because the "relative" underlying uniformity of the material preclude the necessity of 200 sample and averaging : the distribution is more or less flat. On the other hand you cannot do that for a human population with a gaussian or poisson distribution.
But it is not a 200 fix number, you calculate depending on the expected variance how many sample you would need, and IIRC (i could be wrong) that is around in single % or sub percent depending on the distribution and the max population. Think about it as if using the monte carlo method : you need a certain number of sampling from your distribution.
In this case what you are asking for , 200 sample display a thorough ignorance on the statistical significance and the variation expected over the cloth especially since it was examinated. Since the cloth fiber are expected to be from the same time period whatever it is, you do not need in any way shape or form 200 sample.
As for your kip about the scientist "I said the high-tech is thanks to the human brain.
Not created by scientists. " well the scientist came up with the laws of physic, chemistry biology (and maths if extended to mathematician). Not the general population. Thus whatever high tech you are using, is principally because the work of scientist over quantic physic , normal and organic chemistry, semi conductor etc....