I am reminded of when I was young, and smoking was permitted on planes, and there was a smoking and a non-smoking section, but of course it all smelled like used tobacco. (For that matter, I remember when the top of London's double-decker busses was the smoking section, and you'd only go up there at your peril, if it was too full.)
Yes, it is *possible* for COVID to gentle itself. The phase space of "possible COVID viruses" is Vast, using Daniel Dennett's terminology. There's no doubt some huge plateau (stable or metastable area) out in COVID phase space land with a very infectious but almost asymptomatic COVID that also provides robust immune system protection against other variants.
But is a gentled COVID at all likely? Well, I'm only a mathematician, but it has been surmised to me by people who might know better that COVID's spike is both what makes it so infectious, and so destructive, and it would be hard to evolve away from the destructive aspect without losing the infectious aspect which is so evolutionarily desirable.
So probably not.
We're just as likely if not more to see horrible variants. The Devil's Trifecta is of course a very infectious and virulent COVID with a long incubation period and that too would be a plateau in the Vast space of all COVIDs.
The overall solution as you point out would be to reduce the number of mutations by reducing the number of humans who have the disease by vaccinating everyone.
Every day I wake up and remember that if we all got together and organized so most of us remained indoors or at least isolated from other small groups for one month, and the people who did go out were dressed like it was Twelve Monkeys, COVID would end abruptly - or if we made sure everyone was vaccinated and came down hard on defectors (in the Prisoner's Dilemma meaning) and high-quality masking were universal, we'd have the same effect.
We are choosing through our inaction for this to continue. It rots me.