Maybe, maybe not.
The herd immunity thingy does have some things going for it.
A total lockdown early on leaves you with no way to get out of it.
China has that problem. They can't open their borders, because it will let the virus in again.
The virus won't be eliminated. Like measles, it will come back again and again. The more people who have it and recover, the better protected the vulnerable are.

Humanity has only ever managed to eradiate one disease as far as I'm aware, and that was smallpox, and that took an international effort of many years.