To me, asking my T to read a big stack of emails or a journal out of session would feel like an unreasonable request. If it was important to me, I'd expect to use session time. A lot of email and journal content is spent wandering around in the weeds before ever getting to the point.

Expecting a T to read that stuff just seems ... well, when I was in junior high school, before texts and emails, the main way girls communicated in school was in notes secretly passed. Sometimes boys, too. They held great meaning and we'd pass other people's notes back and forth with our closest friends for comment and interpretation. We spent hours at it, largely because we had hours to spend thinking about such things and not too much life outside of school and family. I'd hope my T had enough of a life to say s/he didn't have time for that, but we could talk about it in session. Maybe I'm biased. I'm sure my fave T would have refused a pile of emails. But I sure did enjoy sharing my life that way with girlfriends back when it was the only way we knew how to do it.