> Wouldn't it be great if a vaccinated person could get omicron, have a mild illness, and then have even stronger immunity against other variants?

That would be great, but it appears the immune system does not work that way:

> the body's immune system ... preferentially utilize(s) immunological memory based on a previous infection when a second slightly different version of that foreign pathogen (e.g. a virus or bacterium) is encountered. This leaves the immune system "trapped" by the first response it has made to each antigen, and unable to mount potentially more effective responses during subsequent infections.

https://en.wikipedia.org/wiki/Original_antigenic_sin