You're not really getting rid of negative hit points, you're just adding them to the total and making 0 have more finality to it. Beyond that, this punishes high-con characters significantly more than it should.
High-con characters are the brutes, the tanks, the defenders, the soakers, whatever you call them who are supposed to take all that punishment in place of others, or at least be able to take the up-front punishment that other classes avoid(such as damaging aura effects, localized AOE, ect...)
EX: Joe is a fighter with 30 con. He gets 40HP at level 1.
This means that Joe can only take 11 points of damage before he's "dying", a statein which he hits worse, defends worse, and generally: dies faster.
Fran the mage only has a 14 con, she gets 14HP+4 @ first level. It takes Fran 9 points of damage to enter the "dying" state. The difference between Fran and Joe? 2 points. That's only a 20% difference in damage, while Joe and Fran have a 45% difference in total HP. Joe has almost DOUBLE Fran's HP and yet he can only take 2 more damage before dying.
Not to mention, at first level, a single hit is highly likely to deal 5-10 damage. More on a crit. Especially if you're encountering some bandits who all have longswords for 1d8+2 damage.
If you want a "gritty" system, by all means run one, but I get enough "gritty" in my comics these days and I'd prefer 5e to not be "gritty" by default.