Logistics is all. It's not feasible for so many variables to exist, otherwise the possibility of various frauds is increased.
I understand what you are saying, but I think it's easier to fake the CDC card showing you had both shots than it is one of these printouts from an official clinic. Unless you are really good at formatting anyway, which most people aren't. Also, the clinic is way way easier to verify then a scribbled doctor's name written on the CDC card. All the worker would have to do, is not take your word for it, "google the clinic name and town and state/country, and call the number that pops up, not the number given to them, and verify over phone" Would be relatively easy, and other person could give their permission over phone to release the info, verify DoB over phone, etc.
I wouldn't be surprised if a lot of people are faking the CDC card and getting away with it. I would never do that, plus I do believe the second shot is good for me, even though I personally believe it's more like a third booster for me, I'd rather time it better. But if I have to I will get the second shot now.