Xeviat
Dungeon Mistress, she/her
The real crux of the issue, cutting to the core of the system, is that the mono-stat attack formula makes this possible. If attacks/spells were reliant, even tangentially, to other stats, then this wouldn't be as prevalent. For example, in 4e the rider abilities on a lot of powers were determined by your secondary stat, and as a result I saw very few people go hard on Con in 4e unless they were a tank. If we had spells whose radius was affected by the mage's strength, or a sneak attack that incorporated strength into the damage, then we'd see a lot more people considering those factors rather than just pumping their main stat and then shuffling the leftovers into Con.
I had overlooked this until now. 4E characters did have a strong primary/secondary stat set up. 5E characters lack that to some degree, or at least they lack choices there. For most characters, Dex or Con are automatically secondary stats; Dex 14 is 'important' for characters without heavy armor, and Con 14 seems to be heavily favored. I've seen Primary 16, Dex 14, Con 14 a lot. Only Monks, Paladins, Rangers, and Eldritch Knights/Arcane Tricksters really vary from there.
Could that be helped with more robust subclasses? Maybe the Battle Master could be adjusted to favor mental ability scores? Maybe the Purple Dragon Knight can get more stuff with Cha. Maybe we can get a Tactician that uses Int, or something.
The current house rules I'm looking at end up being a mid to high level nerf for HP in a single combat, but a buff for HP across the day. This may take some pressure off healers to use their spells for healing, and thus would be a damage or utility buff across the board. Since I'm not looking to change the balance of the game with this, just expand players' perceived options, I may need to reevaluate some things. But, it might just make characters more likely to be able to push through 6 to 8 encounters in a day, which I can hardly ever achieve.