Xeviat
Dungeon Mistress, she/her
I'd have no trouble running an 8 CON Rogue, Monk, or any Archer build. Wizards with 4-8 CON are OK too, as I think Raistlin is one of the most interesting PCs in D&D lore.
See, one thing I was thinking of was giving every class the same HD. Heck, maybe just fix it to the size HD like monsters. Mediums get 1d8, Smalls get 1d6. Then, Constitution is THE way to raise your HP. This would need to be coupled with a Con score to HP thing and maybe max HP for PCs or something so it feels right.
I think we'd be in the same situation, but it's an interesting thought. Tanky frontliners would want higher con, backliners wouldn't care as much.
Looking at video games, I know I don't care about my HP stat in games where dodging is an active ability. If I feel good enough about my ability to dodge attacks, I feel like I need HP less. But the way AC works in D&D, heavy armor and high dex light armored characters have about the same AC and nothing feels different between them. If HP were more fixed, than DR could be used more and constitution could be done as Damage Reduction ... but that gets us into a whole nother territory.