Sure, but it's all about getting to the point where it's endemic and no longer epidemic. Infections are the way to overload hospitals and cause unnecessary deaths. Vaccines are the way to minimize hospitalizations and deaths. I think it's abundantly clear one path to getting back to normal is WAY better than the other.
And if there's some way to show empirically that people have the "correct" antibodies (or whatever the right concepts are here) to be immune or have sufficient immunity, then I'm all in favor of testing everyone and counting it that way instead of by vaccine card. But right now we don't have that capability AFAIK. (We can count antibodies, but I haven't seen anything that could translate that into an immunity measure.)
Previously infected by which variant? I'm sure people who had delta probably have a better resistance to delta, but what about the people who had covid last year (i.e. one of the previous variants)? Are they as immune to delta as the vaccine or others who have had delta? There's a lot of variables and natural immunity isn't as easy to gather evidence due to lack of control groups. The vaccines are much better studied and understood simply because the studies are much more controlled.