There's one big difference between software engineers and healthcare: regulation.
A software engineer is hired for their skills (at least ostensibly). No one is required by law to hire someone with a specific degree and specific post-degree training and specific exams.
Contrast this with healthcare. To do certain sorts of procedures, you have to hire a physician. Not because it's demonstrably necessary to have someone with an MD and a residency in such-and-such area do this, but because it's required by law.
As someone else pointed out, this is just the tip of the iceberg. That residency? Residents have no bargaining leverage over their conditions by fiat of residency rules--they cannot leave an abusive residency, for example, to change conditions. Financing the residency itself? Businesses won't cover the expenses because it's not actually worth the costs, so the government foots the bill. And once you leave residency? Well, subspecialty organizations are deciding that it's good to carve out even more regulatory capture with subspecialty credentialing.
People do not grasp how much of this insanity is codified by law and rule, and when they are informed of it, they shrug it off in the name of "safety." It's like terrorism or crime: no one wants to be branded as soft on terrorism or crime, so the government becomes more and more invasive and draconian, and the costs of maintaining the military-police-industrial complex increase and increase. Similarly, no one wants to be soft on safety, so the government becomes more and more invasive and draconian, and the costs of maintaining the medical organization-physician-insurance-industrial complex increase and increase.
There's something disingenuous for physicians to complain about being overworked, and then fight against the things that would alleviate their burden the most: letting perfectly competent professionals with different backgrounds do what they do just as well. But that would mean admitting that you don't need an MD at the apex of healthcare.
To some extent, financial market pressures are doing what I'm saying anyway, as hospitals are realizing that MDs are too expensive as they are. So maybe this is just the first sign of things to come. But the downside of the current system is that administrators aren't allowed to go elsewhere for alternatives, so they just crank up the hours expected of MDs. The upshot is they get devalued without even being given the benefit of being let off the hook.
I guess to address your comment directly: if healthcare were actually a transparent free market, my guess is physician salaries would go down, but their workload would also decrease also. What you'd see instead is much more diversity in who you see for any given service.
The biggest sin of the government in the healthcare debate is willfully ignoring the costs of healthcare, by failing to increase competition, choice, and transparency in pricing. We talk about who pays, but not why we're being charged what we are, and whether or not it's worth it.