Other than the specific "be an archer" from the post I was responding to, I think any class could have those backgrounds. For example even the "enforcer" thug as a wizard could work, you just have to explain it correctly. Instead of flexing muscles, the wizard uses their shocking grasp cantrip (or prestidigitation for that matter) to make their hand crackle with electricity as they reach for the bound captive with an evil grin.
But if the narrative about how a fighter became a fighter is more varied,
that's a good thing. It's just that I believe it's encompassed in the backgrounds and class rarely changes anything other than descriptive fluff.
Or I'm just still missing the point.