Having Con have a strong effect both on hit points and healing surges would be double-dipping and probably not good. But I could easily see a variant that got rid of Con bonus/level hp and had Con add to healing surges instead.
I think that would be a fundamental math alteration that would be unhelpful, esp. as it would impact monsters, so I'd avoid that. In 4E it just felt like it boosted the number of HSes from "I might hit the limit on a very bad day" to "I will definitely never hit the limit", so I don't think it had much real value either way. I think in several years of 4E I saw someone run out of HSes exactly once, and that was in a truly crazy situation the party could easily have rested earlier in.
So if we're going to emulate 4E we might as well improve upon its flaws, imo. CON in 5E is perhaps OP because of HP are so heavily impacted by it, and impacted in a flat way (+2 CON mod on someone with 1d6 HP is huge increase in HP/level, for example), but I think we just have to work around that. I mean, if there's one stat in D&D that could stand to be deleted though, it's CON - it doesn't impact any skills, isn't a primary attribute for any class, and basically only exists for the purposes of saving throws/concentration checks, and boosting your HP into the sky, both of which could be handled differently and likely with a superior overall outcome, but that's a whole other separate thread which I think we've had before.
Class-wise, It think the much higher HP values on "tank-y" classes already address the issue, because HSes provide a flat 1/4 of your HP. So if you have a Barbarian with 120 HP, they get 30 HP per HS spent, whereas the Sorcerer with 60 HP only gets 15 HP per HS spent. I don't believing giving the Barbarian more is helpful unless you also give him a mechanism that encourages him to spend them, but at that point we're talking about a wider redesign (hence my suggesting it solely for Fighters as part of them being able to use Second Wind more often).
I'd also suggest that for simplicity's sake monsters not be able use HSes in combat and give them 3/day instead of 6/day for the very rare occasions they get out of combat.