I think it’s much more likely that doctors will end up sending AI-autocompleted messages that take very little time to complete but seem like took much longer.
The standard will evolve to something like clinic staff sending the messages which will be overseen/supervised by a medical doctor (who won't be involved in composing the messages or spend any time looking at them before or after they are sent). We'll see trials where the doctor who was supposed to be keeping an eye on the messages wasn't, and some flunky kills a patient by suggesting that drinking used engine oil will cure the sniffles. Keep in mind that the messages can be sent from anywhere too, technically no need to keep that onshore. May eventually get legislation that would guarantee that, but only after a wild west period that lasts a few years.
Why would it do that? If they get money for messages sent (rather than messages sent by doctors) then some flunky can send a message and you can still be billed. Given how this sounds like a profit center, and doctors are cost centers (imagine if they needed to send more messages... now they have to pay more doctors!), I figure physicians will be kept far away from any such system.
If we plot out perverse incentives as a space, I believe that our fate will be to trek through every possible point in that landscape such that we eventually see every single one. This means you will eventually get universal healthcare, most likely as the very last of those perverse incentives.