But this grown adult puts themselves in the 'child' position by emailing vulnerable stuff in between the session? The grown adult would go and address stuff in the session, no?

I'm not saying that emailing or not emailing is wrong but just pointing out that if one party for some reason needs to act as a child then why complain that the other party responds as a parent?

That's exactly how transferences are playing out. Transference relationship is not the same as the therapy relationship. Email exchanges happen in the transference relationship while paying for services happens in the therapy relationship