It's not a bayesian versus NHST thing, it's just about doing the right test. You can do this kind of thing easily enough in a NHST kind of way. But yeah I think doing it in a bayesian manner is much easier to interpret and explain, and so requires less training, which is a HUGE boon for the crisis at hand.
You can do the test with NHST only if you know what the researcher had in mind when experimenting. That can lead to absurd results at times, such as your example. With Bayes, as long as no lies are told, your inference doesn't depend on the researcher's private thoughts.
You only need to know how the experiment was performed, same as you do for bayes. If they present the whole dataset, nothing is changed. If they cherrypick and only show the data for the coin that happened to be all heads saying that was their whole experiment, no amount of stats will help you, bayesian or otherwise.
No, if the coins are independent, the Bayesian is not fooled even when seeing only that coin. The argument was in my post above.
The inference in Bayes only depends on the data, and the flips of other coins doesn't make a difference if independent. Frequentist testing can depend on things like stopping rules and hypothesis tested, which aren't correlated to the actual truth and therefore should have zero effect on inference.
They can cherry pick and only show some flips of that coin, but then they really need to be outright lying or you'll ask why only some flips were reported.
