No, they persist in the population, mutating all the while. They spread more when seasonable changes bring people together and when the weather conditions are more conducive to spread.
Covid is a rhinovirus, as are most common colds. From an infection standpoint, I think of Covid as a really bad version of a cold. Colds never go away, they just come back each year with new mutations.
https://en.wikipedia.org/wiki/Rhinovirus
The good news: maybe the upsurge of research on Covid will lead to vaccines for colds and other viruses.
"We’re Not at Endemicity Yet":
https://www.theatlantic.com/health/archive/2021/12/covid-is-...
I was saying covid may become endemic and remain around forever, whereas missedthecue's saying it may be eradicated (which I don't think likely).
But I was dead wrong about covid being a rhinovirus, as yorwba has pointed out below. I've had that wrong for a long time, so need to think about it!