When I flew to AUS week before last, my covid record wasn't checked anywhere at any time, either at LHR or on arrival.

It might have got to that stage where the US needs you to be able to prove you a vaccinated (ie. keep your certificate with you), but isn't proactively checking anymore.

I had similar flying into Ibiza, despite Spain requiring people to be vaccinated. Got checked once in May, but flying into Ibiza June, July, August, September and October, was never checked again. Nor flying in to Barcelona in August.

I think as long as you can prove on demand, you'll be ok.