There is another explanation for mental powers in the original trilogy: they are a result of Second Foundationers being experts in psychology. Essentially, the idea is that if you are a really good psychologist, trained since an early age, you are pretty much telepathic. And psychohistory in the original trilogy is seen as a development of psychology applied to large groups, that's where the name comes from. So the two are connected originally.
That doesn't contradict experimental treatments or the genetic angle. It's perfectly possible that being good in the psychology department can be enhanced with some treatment and depends on genes.
But in the TV series Salvor Hardin can predict coin tosses, and that isn't something you can achieve by being a really good psychologist.
I see two possibilities to keep it compatible with the original idea:
Gaal, who is telling us the Foundation story, isn't telling us the whole truth. The coin toss story is in place of some other event similar to a coin toss. Salvor correctly predicted something that most people would think was pure chance.
Salvor is claiming that she can predict coin tosses just to cause an impression, but she's just using a loaded coin and it has nothing to do with her mental powers.
There is another explanation for mental powers in the original trilogy: they are a result of Second Foundationers being experts in psychology. Essentially, the idea is that if you are a really good psychologist, trained since an early age, you are pretty much telepathic. And psychohistory in the original trilogy is seen as a development of psychology applied to large groups, that's where the name comes from. So the two are connected originally.
That doesn't contradict experimental treatments or the genetic angle. It's perfectly possible that being good in the psychology department can be enhanced with some treatment and depends on genes.
But in the TV series Salvor Hardin can predict coin tosses, and that isn't something you can achieve by being a really good psychologist.
I see two possibilities to keep it compatible with the original idea:
Gaal, who is telling us the Foundation story, isn't telling us the whole truth. The coin toss story is in place of some other event similar to a coin toss. Salvor correctly predicted something that most people would think was pure chance.
Salvor is claiming that she can predict coin tosses just to cause an impression, but she's just using a loaded coin and it has nothing to do with her mental powers.