I dont see how something like a name is unique enough.
Sure you could use an email address but that is often more complicated than a phone number.
Its all well and good for personal use but for business use you need to be able to easily identify someone and number are easier to give out over a phone than alpha characters.
Me and the wife tried out Facebook Messenger over a Edge connection, the voice calling part. Was actually pretty darn good quality with minimal delay. If the phone option on Edge does not work there are always voice messages.