The moral hazard of legitimizing natural immunity is very real. Yes the biological science shows that natural antibodies work well, but social science tells us what dumb people will do it that's supported as a solution.
It hasn’t for other vaccines/diseases. For example when I went back to grad school and didn’t have a shot record they they tested me for anitibodies instead of using more vaccines. I was low on MMR so got a booster but didn’t need varicella and various others. No big deal.
That need you to be infected badly enough your body created the long-term ones (so not complete asymptomatic), and also not badly enough you die or get long covid (which by the way some measure at 50 % infected having in some form - but it's hard to say especially since long covid is not well defined).
Which while might theoretically be true for individuals, it is too unreliable to make a policy of it.
EDIT: but yes, you're right, people (whether they are bad actors spreading misinformation willingly or stupid plague rats believing them) will catch on that and claim they don't need the vaccine then.
Just like they caught on "vaccinated people can still get covid!" - sure they can, it is just two orders of magnitude less likely.
People are just saying (and science confirms) that previously infected shouldn't be discriminated against or forced to vaccinate.