Isn't that what happens now with normal exposure to the virus?

No, because for herd immunity to set in, far too few people have SARS-CoV-2 antibodies simultaneously (even if we let COVID-19 run rampant). That, in a nutshell, is the main reason why the “herd immunity”-based mitigation strategies are being dismissed by the vast majority of immunologists and epidemiologists. In fact, “natural” herd immunity (without the help of vaccines) might never have eradicated a single disease — at best, it leads to a temporary reduction in numbers.

The “standard example” for natural herd immunity eradicating a disease is the black plague, but most experts now believe that herd immunity isn’t actually responsible for its disappearance from Europe.

At huge societal cost. The virus has already killed almost 250,000 people in the United States.

Vaccines get you the individual protection (and eventually herd immunity) without all the death.

Right. We have about 3% of the US population who has had COVID, and 235,000 deaths. To get to herd immunity, we need about 50% of the population to be immune (exact numbers vary depending on the study). So to get there with the virus, we would expect around 4 million people to die in the US alone.

Not if you can get re-infected. That's why the cold and flu virus are still doing the rounds every year despite pretty much everyone you know having had a cold or flu at some point.