To be clear, I believe at least that section of Taft-Hartley is arguably unconstitutional, and worthy of getting civil disobedienced hard. I just don't see it as likely to happen in such a way as to be effective in facilitating change. Add on top of that the onerous burden of medical school debt, and I don't exactly see many being willing to make that choice.
Like sure, they aren't solely to blame, and it would probably be hard for individual doctors to successfully push for systematic changes, but the flip side to that is that they don't get to say "Oopsie doopsie, it sure isn't great".
After all, a care facility can probably be reasonably confident that if a patient needs help, doctors will feel compelled to care for them.
Put another way, would you want to walk into a hospital/ER/your doctors office for something acute and get an apology that care is unavailable because the doctors are on strike?