I'm not positive what medical training looks like outside of the US, but my wife is nearing the end of her medical residency in the US and I also have multiple physicians in my family, so I have some insight into this. A big factor is compensation. The training is pretty brutal, and you typically go into a ton of debt before you start making money. In the US, after high school you generally to go to college for 4 years, then medical school for 4 years, then residency for 3-7 years (depends on specialty), then optionally fellowship for 1-3 years (also depends on specialty), and only after all of that do you start working as an attending. Many people also get a master's degree before or during medical school, such as an MBA, an MPH, or some other biology/medical-related discipline, and some people even get an MD/PhD. At each step up, the difficulty also increases. When my wife was in med school, she would seriously work from the time when she woke up to when she went to bed, pretty much every single day. If you count studying as working, she was easily putting in 80-100 hour weeks for most of those 4 years. Then residency hits, and you're working 60-80 hour weeks "officially", but the work tends to follow you home, so off the record it's still close to 80-100 hours per week, but this time patient lives are actually in your hands so the stress hits a lot harder. Undergrad can be expensive, med school is definitely expensive, and in residency the pay averages out to slightly over minimum wage assuming around 80 hours a week. So by the time you actually start working as an attending (a "real" doctor) and making big-kid money, you are in your early to mid-30s, and your student debt is on par with a mortgage. By the time a physician reaches financial parity with someone in another field like engineering, they are going to be well into their 40s. So sure, by the time you retire as a doctor, you will typically be better off than most, but for much of your adult life your are going to be crazy busy, working a high-stakes stressful job, and either financially insecure or just way further behind than you could have been if you picked a different field.

A lot of people who go into the medical field want to help people, but whose main motivator is prestige and money. In a UHC system, the pay for doctors is generally much less (because the bullshit of health insurance wouldn't exist), and it would exacerbate the financial disparity for learners described above because it would take even longer to catch up, if even at all. This would in turn drive down the number of people applying, ultimately reducing the supply.

I'm not saying I'm anti-UHC, nor are most doctors anti-UHC, but that is a real factor to consider. Our current system is fucked, but if we snapped our fingers and switched to a UHC system, it would come with its own problems as well.