"Is it a myth or is it true that most mental health professionals like psychiatrists and psychologists have therapists of their own, just to help them keep on track after dealing with patients with mental issues all day, every day?"
It used to be standard practice for a full fledged therapist to have a consultant. Many still do, but some do not. Of course, when they first start out, in many states, they all go through thousands (literally) of hours of supervised practice with regular sessions with their supervisor / consultant, who is a therapist with more experience. They earn something like 1/2 of their supervised hours before getting their degree, and the other half after they get their degree, but before they are licensed. In California, for instance, this is true of people practicing psychotherapy who are psychiatrists with an MD, clinical psychologists with a Ph.D. or PsyD and a psychology license, and therapists / counselors with master's degrees and licenses like MFT, MFCC or LCSW (terms used in California and some other places).
Virtually all, but I have known exceptions, have gone through their own therapy as a client*
Some states are more lax than others. In the 1950s, in some states, a person could hang out a shingle without a license. That still happens, but there are usually state laws that prevent them from using titles like "therapist," unless there an adjective (e.g., past lives therapist) that nullifies the impression that they are mainstream.
I don't know what the rules are in Texas or in the military.
As far as it being a good profession to be in, yes, but ... MDs in psychiatry have a rather high suicide rate. I used to be connected to a training clinic for future MFTs and future Ph.D.s and PsyDs. One night, long after hours, when a supervisor who had to drive in to help an intern with a client in crisis, as she was walking to her car said, "Why does everybody think this is such a neat field to go into?"
*Back in Freud's day, this was known as the "training analysis," and was required. It recalls the advice of another person some 2,000 years ago: "Before taking the sawdust out of your neighbor's eye, take the great plank out of your own."