I agree that OP's T has been clear about the limit/boundary here and OP continuing to try to get a response is futile. I question (or take issue with) the use of "belief" here as I wonder if it is a belief that the OP will be able to get a response or if it is a need to have the support of out of session contact. If it is the need to have that type of support/relationship with the T, then perhaps this T is not the T for the OP (something that has been commented on before). I believe, this T has stated that they do not do supportive therapy. It does seem like an impasse.

Then again, like all comments, this is colored by my own issues. I see this type of situation very black and white. The other person says no *and* if it is something I feel strongly about; I can't make them do it, so I leave and find someone else that will.

So OP - how badly do you need out of session contact or support in your therapy? What would those things do for you that will help your treatment? What did they do for you then? Are there other ways you could still get that now? It sounded like to me that you were using emails as a method to share information in a less personal way verse what might be considered object consistency seeking behavior. Is that true? If so, does acknowledging that within yourself change how you see his decision and perhaps finding a way to work with it.

How much is your treatment being hindered by not getting them? What is the price you are paying or the cost of not getting something you want (and maybe need)? What about this is the bigger/root issue and can you get those needs met differently within the context of what your T is willing to do? Can you grieve the loss (I do see it as a loss)? Can you separate out the grieving process from what is a therapeutic need?

I think it's perfectly fine to continue to talk to him about this because having something taken from you is a big deal and comes with lots of baggage. BTW, I still go back to the concept that you could call him if you needed support; unless that is also off the table. I know for me taking that step to call is a major leap where as emailing is much easier. Even with that, there have been a few times where things have been so bad that I did call her.