I am not a priest but I have a couple of thoughts on the matter
1.) I was visiting a religious order and they gave me a little book called the St. Ignatius Daily Examination of Conscience. In this book, one of the things you are supposed to be reflecting on at night was whether you had answered all your e-mails or not. I guess it could be considered the same as avoiding phone calls. It seemed odd to me and obviously, someone had revised it to apply to modern day society/culture.
2.) A lot of priests are in the age range where they are uncomfortable with e-mail.
3.) There are huge confidentiality issues with sending an e-mail to a priest. Your e-mail is always stored on a server that someone other than the priest and yourself has access to. In theory, that “someone” should have signed a confidentiality agreement but you still have a third party who has access to the information you send.
4.) From my own personal perspective, I read all my e-mail. But if I receive an e-mail message on my cell phone, I only read it with the intention of making certain it is nothing urgent. I hate to respond to e-mail on my cell phone because it is enormously time consuming. Then when I get home at the end of the night, my intention is to respond to the e-mail, but I frequently get caught up in other things.