I used to have this doctor who, frankly, I only saw once. But I liked him a lot! He treated a foot injury with grace and good humor and a couple of years later when I had a problem with my other foot I was eager to go see him again. But it turned out I couldn't. I'd switched jobs in the interim, and the new job's insurance plan didn't cover that doctor.
Except even though it's factually true that I saw the first doctor before the Affordable Care Act passed and was unable to see him again after the Affordable Care Act passed, obviously the issue here had nothing to do with the Affordable Care Act. It had to do with, I dunno, human freedom? The vicissitudes of life? It would be genuinely insane for congress to try to pass a law that would prevent the occurrence of a situation in which a person's insurance plan ceases to cover visits to a particular doctor. How would it possibly work? Once again, if people want to slap the president for overpromising they should feel free. It seemed clear to me in context that what Obama was saying at the time was something like "HHS bureaucrats won't be picking your doctor for you" rather than an unrealistic promise to enslave medical professionals and force them to keep treating you indefinitely. But nitpicking is fair in politics.
But on policy, this isn't even close. What would a legislative guarantee that you can keep your doctor look like? What if your doctor is really great and half the people in town want to see him? Some kind of system, whether it's rationing or high prices or whatever, is going to have to ensure that only some people get in. That's life.