Honestly, while it could work from a game balance perspective, I'm not really the right audience for that sort of thing. I've never been down with anyone being able to substitute one stat for another, in any circumstances.
Not a fan of Hexblades/Shillelagh/Battlesmiths, huh?
Every other stat has at least two classes based around it, but Constitution has none, because of the wonky "balance" between the ability scores in 5e. If you want to truly balance all the ability scores in the game, as well as making hit points have a bit of a refined definition, it's probably a good idea to make hit points use the ability scores your class focuses around and have there be classes/subclasses that focus on using Constitution.
That's just because Dexterity is broken, and does too much. If Constitution only gave as much benefit as Wisdom, it could be dumped just as safely.
(fixed a type for you)
Yes, Dexterity is unbalanced compared to other ability scores. Most characters tend to have higher Dexterity than Constitution, in my experience.
Also, a paladin may think that they can dump Wisdom safely, until you're ambushed by bandits or fall into an obscured pit trap. Ability scores should have a major passive use and major situational use, IMHO. Dexterity is passively used for Armor Class, but situationally used for Initiative. Strength is passively used for carrying capacity, but situationally used for shoving/grappling. Constitution only gives the passive benefit of hit points (which I think should be changed) and its situational uses are minor ones that almost every other ability score shares; Constitution saving throws, holding your breath, and the very rare Athletics (Con) checks for swimming.
If they were to try and rebalance the ability scores, they would have to give CON major passive and situational uses. Passive ones could be hit points for certain classes, an Exhaustion buffer (adding the amount of Exhaustion you can take before you get negative consequences from it), and other uses (my mind is blank at the moment). Situational ones could be an Endurance skill, an increased amount of Concentration abilities, an increased amount of non-spell Constitution saves, and being used for Death Saving Throws (which need a bit of tweaking too, IMO).