I do have mostly positive experiences of outside contact with my T. We've taken a while to find out what works for us in terms of outside contact and we are both better at managing it in a way which ends up being helpful about 90-95% of the time.
There's definitely been a shift in the way he responds to my emails over the years. Initially he would either not respond, or respond with a brief email acknowledging receipt (which is what his contract suggests he will do) but now he usually tries to respond to my email in a meaningful way, and it rarely takes him longer than 24h to respond. I think the trust has grown mutually. He always says he trusts me to make decisions on extra time, extra sessions, etc, and I suppose, like your T that extends to regulating the necessary out of session contact too. If I don't want a response I tell him so, and on those occasions he won't. It really seems to work for us and I find it a helpful part of therapy.