A valid payment method has to be accepted. If the TOC is unable to accept a valid payment method, it becomes the TOC's problem. Travel cannot be denied for this reason.
If TOCs no longer wish to accept American Express, they would need to remove it from the list of valid payment methods.
In the real world, as this example shows, if TOCs no longer wish to accept AMEX they just tell customers one day that they don't and nothing comes of it.
From memory not all TOCs accepted AMEX, I think Virgin never accepted it. Might be wrong, but AMEX was never a mandatory method of payment.
I believe this will have been the case at one point yes. However, it is now listed and therefore they MUST accept it (in theory, although this thread shows that they do as they wish as per).
As an AMEX user i do try to use it where possible, it offers me the best additional benefits for doing so so i'd be daft not to. However, i am perfectly aware of its issues and i'm not the sort of person that would go around with only 1 payment method on me anyway, but even if i did it certainly wouldn't be AMEX. I appreciate that everyone's situation is different and it is not for the railway to expect a customer to use a different payment method IF they have a working one that is listed.
In terms of tvm usage, it's interesting seeing the talk about contactless not working but chip and pin working. I haven't used a TVM since early last year due to the better availability of e-tickets since the world woke up again but certainly at that point (so notably out of date) the Northern TVMs would not accept AMEX using contactless OR chip and pin but i'll certainly try going for chip and pin next time i use one. Hopefully this has improved.