Except this doesn't happen, otherwise COVID would have already stopped spreading. Our social response to COVID isn't adequate enough to stop its spread. We can't get people to actually lock down or stop transmission.
When everybody goes inside, all the variants stop spreading. The pressure is the same on every variants, so there's no source of evolutionary advantages on any variants.
Any variant that is able to co-exist with (rather than kill) its host during a full lockdown has a significant advantage. We co-exist with several coronaviruses already.
But it is — this will not be the response to a variant that doesn’t cause an increase in hospitalizations or deaths.
A given response exerts pressure on all variants, of course. But a milder variant may be able to spread widely before a more lethal one shows up and ratchets up lockdowns.
When a more severe variant starts dominating and everyone goes inside and stops spreading it that makes the more severe variant die out. As it dies out, and people go outside, variants that are milder will thrive because people won't lockdown as easily. This can repeat forever until eventually the virus is completely benign. Of course it's all based on randomness so it's not exactly predictable what will happen.