I have not heard of any counselling done via email. I have been with my Psychiatrist using telehealth, talking on the phone. Because Dr is a private practitioner, I do pay, and Medicare pays too.
I have had reason to consult a Psychologist, and asked my GP, who wrote up a Mental Care Plan. I did not have to pay. It was not ideal, though, because of the limitted number of sessions, and I could not get regular appointments. We did, a couple times, have the session via phone.
Telehealth has been a good alternative during lockdowns - reducing the need to go out, reducing risk all round.
Long ago, when I first began seeing a Psychiatrist, I could barely manage to open my mouth, to say anything, and it was a couple years before I felt able to open up a little. In the meantime, I wrote, wrote, and wrote. And painted too. I showed some of writing and paintings to them, and slowly began to talk.
It was as if I needed to push myself, stick a wedge (like the writing and painting) in there and create a crack and keep opening that up, learning as I did, I could trust the Psychiatrist to hear me. That was incredibly important to me.
Also, I should mention, talking to my current Psychiatrist over the phone is not ideal, for either of us. It is much better getting up and out to a seperate place for the sessions. Challenging, but better.
If we were talking via email, there would be a lack of spontaneity. Not the same interpersonal interaction that is available even over the phone, There is more face to face. Some people do that, you know, Face Time, Zoom, but not me. I cannot hardly see when we are together in a room, much less on my phone's screen. So if Dr can't see, I think that levels the playing field!
Most people do take advantage of non-verbal cues, body language and such when talking to others, so if you remove those from your conversation, problems may arise - miscommunication, not seeing or hearing the emotional content, when words do not convey the full complexity of what we express.
I hope this is helpful.
Even as you write on this forum, I think you may find words to say will come more easily. Be patient. Sometimes, when I pushed myself, frustrated and angry with myself, that was not helpful, so "patience", I say, "patience", like a mantra.