> they select for the spike protein of the original variant and ignore others
Repeated exposure causes you to get many different kinds of antibodies that react against many things like the original spike protein. There's ample research showing that a 2 shot (and 3 shot) series of mRNA vaccination creates a very broad response that cross-reacts to many spike protein variants.
Reaction against the spike protein is best, because this binding doesn't just activate the rest of the immune system, but it's neutralizing against the virus: it prevents its entry into cells.
Compare this figure:
https://cdn.jamanetwork.com/ama/content_public/journal/jama/...
Many convalescent (prior infected) individuals had very low neutralization activity against variants. All vaccinated individuals had good neutralization titers against all variants measured in this study, though they were still definitely better against A.1 than other types.
> whereas natural infection generates a broader response that will target a variety of proteins.
Natural infection will generate a variety of nucleocapsid antibodies, too (against non-spike portions of the virus).
This is a mixed bag: it means the immune system can be activated and it also means that the immune system will try and break down things that have antibodies bound elsewhere on the virus... but the spike antibodies around that could directly prevent infection of new cells are less effective against variants.