The issue is that even though there is no shortage in the US there is a shortage in the rest of the world. WHO is begging wealthy nations to not boost their vaccinated and instead share with nations that need the vaccines. Should we just ignore the rest of the world and just boost every 6 month? Someone might decide not to get vaccinated after recovery for the same reason you wouldn't buy 2 lottery tickets instead of 1 when deciding to play, your odds of winning are much higher, but they are still so tiny that not playing at all is a better decision. People play anyway, but with covid we are supposed to follow the science not just do what makes us feel better regardless of evidence. If there were no shortages of vaccines everywhere then sure vaccinate as often as you want.
In the US you could prove recovery by showing positive and then negative tests or by showing a test indicating you have antibodies, but these are not taken into consideration. CDC basically used the discussed study to say that since protection goes up when vaccinating recovered then just vaccinate everyone because there is no harm. This is true, but it ignores the rest of the world that is lacking in the vaccines. Sure CDC maybe is not supposed to care about the rest of the world, but the pandemic is global, so solutions need to be somewhat global.
In any case at this point covid discussions have nothing to do with science and are more like religious believes. Some believe vaccines are evil, some that they will solve everything. Add to this all the political baggage and it is very difficult for a regular person to make the right decisions. To me it is pretty clear that unvaccinated should vaccinate even more so for vulnerable, recovered I don't know maybe at some point we need more studies on how long natural immunity lasts, I also don't know about boosters. Especially boosters for recovered + vaccinated, I am sure the booster will increase protection even more. I also realize that choices on individual level of either recovered or vaccinated will not help the world anyway. It is not as if I forgo the booster it will go to some other country to someone who needs it more.
1st point - showing valid proof : Sadly, in the US, everything is non-standardized. Meaning that test results are unique to a particular lab, insurer, drugstore, whatever. There is no standardized reporting format nationwide, so there is no standardized way to validate them. In Europe, for all the faults we have here, we DO have that in place.
2nd point - the numbers aren't lottery ticket numbers. This is true of other diseases as well: having a light case might mean your body never developed a strong immune response, so you might not have great protection despite having had the virus. Also, there is plenty of anecdotal "proof" that protection wanes over time - mostly in that nobody gets the bug again within 6 months of having it (literally, nobody) but people DO contract it after. And any time it's active in a system, even if it's a mild case, it's an opportunity to both spread the disease AND to develop a variant. Being vax'ed is very solid protection and being vax'ed plus victimized is even better. Everything agrees on this. I don't need to know precise numbers to say that it's a good idea to get vax'ed. I guarantee that 99% of people out there have no idea how effective the polio vaccine is (Salk's was only about 65% effective), but damn near everyone had it and because of that, polio was defeated in most of the world. Chicken pox has basically been defeated in western countries.
I agree that the difference may or may not be marginal - but there IS a definite difference and, given the number of people who are screwing it up for everyone else by not getting vaccinated, having everyone else who doesn't hate society and doesn't hate their fellow humans taking an extra, free, quick, nearly painless step to better protect all of us seems like a good idea.
3rd - Yes, you not having a vaccine or a booster will definitely not result in that vaccine being administered in Rwanda. We'd need teleporters, we don't have them yet. I wish we did. There's a disappointing and sad side to that; on the other hand, imagine the wrath and invective that would be hurled at the administration in France, the Netherlands or the US if people who wanted a vaccine couldn't get a vaccine. So the people in office, depending on our suffrage to remain there, have it in their best interest to have vax's available.
Note, I am pretty much agreeing with you on most of what you say about the hard-ish science, I just think that it's important to avoid excuses for not getting vaccinated, and it's important to put the context out, well, at least my view of it. I have nothing against you and I'm not "arguing" as much as looking at the point from a different direction.