This was a possibility at some point, but it required a very high number of people to get vaccinated quickly, before more variants. That did not happen, so the newer variants the vaccines had lower efficiency, which required even more people to be vaccinated, which is now seen as impossible, and so instead of hoping to get rid of Covid, now we hope to make it endemic, which is also achieved through more immunity, that which can come from either surviving Covid, or a vaccine. In the endemic state, most everyone has had immunity through a vaccine or recovering from Covid at least once. Covid is still around, since it evolves quickly to bypass immunity and spreads too fast to be eradicated, but as something that people get say yearly, or at a rate more similar to a cold or the flu, you get in this sort of balance where you're always immune enough to survive the next one and evolve your immunity along the way.

The issue is getting to this state with the least casualty, and vaccines are best for that.

I checked for other pandemics to see how they ended, and they all seem to end the same way, by reaching large herd immunity either through vaccination, or just everyone catching it and having the weak die off, and the survivors pass on immunity to the next generation.