I've also never heard of your immune system targetting a genome; how is it generally gaining access to that genome when it is encapsulated?
My understanding of the spike targeting is that COVID is only so infectious because of the specific configuration of that spike protein. Change it too much, and it stops being nearly as infectious (and thus stops being COVID as we know it).
Why do you think that?
A genome codes for protein production. T-cells are something like a Unix grep for protein sequences (instead of character regexes) with their grep argument a memorized protein sequence to recognize in the wild.
So it's some sort of dumb luck if the entirety of proteins, specified in a genome's blueprints, would happen to have t-cells assigned to recognize all of them.