About 20 years ago I had a phone number that was one digit off from a local attorney who took medical malpractice cases. And despite not using it myself, that phone line had built-in fax detection and so it would accept any incoming fax and then email the results to me. I lost count of the number of times I got sent a good chunk of someone's private medical records. In many cases repeatedly from the same source, despite me calling them and politely explaining that they had just sent someone's medical records to a random stranger.

So ... perhaps more secure than email, but not necessarily as much as you'd think. Though maybe it isn't as common now to have phone lines with fax receiving included. I worked for a CLEC at the time and got my phone & internet through them too, so maybe I was just getting the special deluxe plan. Had my very own public class C, too, but that's a different problem.

As if phone lines are not inherently insecure. The only reason they use fax is because it's been grandfathered in.

Are you sure? I work across a few hospitals in the UK and can't recall seeing a fax in the last decade.