Sure, a person may play more cautiously, but this is purely strategy-based. It isn't very interesting narratively or from a role-play perspective.

HP isn't supposed to be. HP is supposed to facilitate combat so the DM can handle the narrative. In fact, if HP had a narrative component like giving people black eyes or other wounds, that would be the first thing I would discard. Mechanics are dispassionate and incapable of feeling out the story, so when it is narrative-ly appropriate to tell the mechanics to shut the hell up and take a sideline, they don't. That's my job, and HP does a goddamn glorious job of keeping combat going without stepping on my toes as Storyteller.

So let's say I've got two choices. A system that will dictate when players get a black eye, and requires me to intervene if I don't think it works with the narrative, whether it's anti-climactic, or an unlucky roll that doesn't feel particularly interesting. And if my players know the rules with any degree of real familiarity, I now get to deal with them wondering why I bent the rules to avoid giving them a penalty.

Or a system that helps them understand their potential to keep going, regardless of whether or not they're fighting or even how they're fighting or dealing with anything else that can drain HP. Then, it's my prerogative as DM to decide when someone takes a black eye or any other "non-HP" wound (or hell, even whether or not they end up covered in blood). It lets me decide to protect them from extra wounds when the fight is supposed to be easy, to show off how powerful they are. Or it lets me make sure half the party gets black eyes from the bar fight. And I can do this despite both groups of enemies having the exact same stat block. Because the stats are meant to facilitate the game, and get the hell out of my way, so I can weave an interesting narrative.

The only time I've ever played a D&D game that had rules for wounds, everything from cracked bones to dismemberment, how to determine them randomly from critical hits, how to target enemy body parts, how to defend against such attacks, the penalties for suffering such wounds, how to recover for them, the works, it became the one and only D&D book I've banned from all my games forever and ever until the end of time. The end result is that things that should be interesting become commonplace, and suddenly aren't interesting anymore.

The fact is, mechanics can't tell a narrative for shit. They can work with the narrative, and weave with the narrative, but when the narrative needs them to sit down and shut up, they don't know any better.