I was reading about this on quora where actual therapists reply. Many will not outright say it as a boundary. It can seem more personal and effect clients differently. I think it's not that common for them to outright say it. Some may. Some may not.
This t cares though. He's proved it in many ways and saying he cares about your well being is a way of saying he cares about you. Not every t will have the same boundaries but all of them have them. This is his. He should not have to say it if he has that boundary. Just know he isn't outright telling other clients either

I am just not sure even hearing it would make you believe he cares. I think the uncertainty in his style causes a lot of doubt for you. Understandably. A new t with attachment experience might be great for you. Your call of course, just suggestion