No, first of all due to imperfections in the manufacture of real coins, there are actually no fair coins. Also the bias in the probability affects the first throw as well as all the rest. If your dataset is composed of first throws/rest of the throws, you’re going to see they are correlated.

I think you're missing the fact that you don't have to chain coin flips literally right after another.

As the other commenter said, in between coin flips, use a highly secure PRNG to orient the coin randomly. This would correct for your bias (if true).

You're missing the point.

A coin that is biased towards heads is one that would more often land on heads regardless of how you hold it when you start the flip.

The study finding is that every coin is more likely to land on heads if you start it with heads facing up, and will also be more likely to land on tails, if you start it that way instead. This bias, while small, is greater than the typical observed bias due to imperfections in manufacturing.

It's not about the "first throw" vs the "rest of the throws". It's about how you hold the coin when you go to flip it. That's what they mean by "started".