No, there are too many variants to achieve any sort of herd immunity.There can’t be many people left to get it. Are we reaching a sort of herd immunity?
Probably 90% of people in my extended social group who are out and about all the time have had it but probably only have my extended family, who are older and more sheltered within the community.
NZ has just had a BA.2 wave, with BA.4 and BA.5 now taking hold. BA.2 infection immunity doesn't hold much weight against the latter variants (links below)
If you've had BA.2, there isn't much immunity to the later variants.