Doctors can’t be bothered doing anything differently, and they often don’t work for the place they are providing service at, and don’t have reliable access to email.
As doctors get swallowed up by medical networks/cartels, fax will shift to EMRs and patient portals that will leave you wishing for a fax machine.
Why? It would be very unusual for a doctor in a wealthy country not to have a smartphone with service, or the ability to get one.
You also have the scenario of unaffiliated doctors... your GP doesn’t work with your urologist, and they do not have a secure way to communicate.
It is possible to use things like Zix or other “secure” email solutions, but all are painful and all will vary from organization to organization.
Nothing about a smartphone prevents email from being encrypted. The fact that nobody's sending encrypted email does, but that's a user adoption problem rather than a technical problem. The technical problem is solved, solved well, and has been solved for decades.
The world needs something like iMessage but more open.
We've had the ability to do this for a long time, but only a few major players are in a position to ensure sufficient user adoption, and they're not interested in creating anything that doesn't drive users to their walled gardens.