I wonder if charging for these messages will make doctors more likely to respond. I wouldn't mind a modest copay for these. But if I pay $30 for an appointment, a quick message should be under $15. There would also have to be some sort of SLA, or the ability for a patient to specify that a question/email has expired and no longer requires an answer.
I've had situations where I waited for a doctor's callback so long that I ended up calling a friend who was a physician assistant, watching a youtube video about how to fix the problem, and taking care of it myself. The only way that process could be any worse is if I got a reply after I had fixed it — and a bill for the message.