And yes, standardizing such testing, as complicated and expensive as it is, across all the relevant labs across the country would be a bureaucratic nightmare.
I'm not going to engage with your bullshit about doctors faking vaccinations.
Fake vaccinations[3], including fake certificates are popular, especially when they are forced. If you want a fake vaccination, I know a guy who knows a guy. Point is - it's not bullshit.
Yes, checking certificates is easier than testing for antibodies. While harder, I still think antibody tests would have less false positives and I most certainly don't think the relative easiness to administer vaccines is a good reason to require everyone to take them, even when they are more immune then those who only got the vaccine.
A friend had an antibody test as he already had covid and didn't want to go for a vaccine. It was no big deal.
[1] https://www.covid.is/sub-categories/travel-to-and-within-ice...
[2] https://www.landlaeknir.is/um-embaettid/greinar/grein/item43...
I don't even know how Iceland's health system is organized. But it would be a good bet that it is light years ahead of the mess of the 52 states plus territories and whatever! Much different story.
> how complicated the testing is or how many health providers and labs there are.
AFAIK, you can actually test blood reactivity to virus protein, proving a positive.
I'm not sure how you would prove that a person actually got vaccinated and it wasn't the case of doctor pouring vaccine down the drain, someone else taking the vaccine instead etc.