Video games are never going to be free of health systems. It's just an inevitable necessity of the medium. A game is a challenge, and a challenge will have a victory and a failure state. If we want to avoid the failure state, there needs to be something that indicates how close we are to succumbing to it. And usually that's a health system. They've been with us since the beginning. They used to be called "lives."
No, health systems are probably never going to be realistic, because in a truly realistic game you wouldn't be able to continue after getting shot once, and any game that doesn't let you make any mistakes isn't going to be much fun for anyone except insane obsessive no-hit-runners. If you're shot in a non-vital area, you could just disable the character, slowing movement speed or reducing their ability to use weapons, but there's a reason why many retro games with lives systems make you temporarily invincible after being hit. When you're in a bad place and you make a mistake, you're probably going to keep making mistakes if you aren't given the edge to immediately get out of the bad place, and you certainly are going to keep making mistakes if you've been hobbled.
So yes, surviving having your flesh mangled by bullets is terribly unrealistic. Personally I've always thought, in a realistic game, health should be replaced by a "luck" system. When the game calculates that a bullet is about to hit you, it corrects the trajectory so it doesn't, and you lose a bit of your luck bar. Then when it runs out you finally get hit by a single bullet and go down crying like a big fat girl.
What's not a great health system is health that swiftly regenerates by itself after you've not been hit for a few seconds. And it's not just because it's unrealistic, nothing's going to be realistic, we've established that. It's just really boring. It's a cut-down experience.
Before the health regeneration glut in shooters, health was a resource you had to manage carefully. It added a new element to the battlefield - if your health was low all you could do was try to sprint to some kind of restorative before you succumbed. And if there wasn't a source of health nearby, you just had to be extra careful. And weren't those the most exciting parts of a game like Half-Life, when you've scraped through an encounter with several organs missing and your arms hanging off so now every health point counts and your wit must be about you at all times?
I do not know how anyone could have thought that regenerating health would enrich that experience. When your health can regenerate, all you need to do is, as ever, hide behind a wall and wait for a bit.
What regenerating health also does is effectively reset the game between every fight. Nothing is carried over, it's just a sequence of unconnected and yet eerily similar gunfights. Without health restoratives, you're also limiting the means by which the player can be rewarded for exploring or finding secrets. All you can offer is more ammo, which they can pick up all the time from fallen enemies anyway.