This is just pointless semantics. It's a pure coincidence that Bayes theorem (which is a simple set-theorectic result) shares a name with Bayesian statistical methods.but if they use it for statistical inference in such a manner that Bayes theorem replaces the frequentist definition of probability then they are de facto doing Bayesian statistics while merely pretending not to.
If Bayes' theorem had been called the law of equal areas, we wouldn't even be having this argument.
Even if we accept that Bayes theorem is part of Bayesian statistics, then the debate is simply between "type A statistical methods and "type B" statistical methods.
But, fundamentally, you cannot simply remove a key theorem from a mathematical structure. It's a bit like trying to commandeer the quadratic formula and saying: you can do your mathematics but you can't use the quadratic formula.
You can prove the quadratic formula from the axioms of algebra; and you can prove Bayes theorem from the axioms of set theory. You cannot just remove a theorem. Even if you try to take it away, what do you do when I prove it again the next day?
How do you stop me using Bayes' theorem, even if I never call it by name and never write it down explicitly? I can just allow the rules of set theory to do the work. Just like I could without ever using the quadratic formula. It would just happen in the background.
In fact, I like using the probability tree method. Bayes' theorem does in fact fall out of that and usually I haven't used it explicity.
This is absurd!