So, I had abut 20 years of (mostly unarmed) martial arts (Kenpo, Boxing, Muay Thai, Tae Kwon Do, some other stuff) before I took up sword-fighting. And one of the absolute shocking realizations to me about the latter is that the style of fighting is very different.
When fighting with a sword (especially out of armor), you make damn sure you don't frakking get hit. So you make very different trade-offs in combat. Getting hit by one blow in 10 gets you not banged up a little, but dead, or at least at risk of it. Fantasy heroes could probably take way more blows than would be appropriate in reality, but I have no problem with hit points mostly turning hits into light scratches, near misses, and the like. To me, after a fight, your typical D&D party is banged up about like Indiana Jones. Give them a bucket of hot water, some bandages and a little whiskey or vodka and they're basically right as rain, because they were never that hurt.
"Bloodied" (which I too miss, by the way) would be like when Indy actually gets shot in the arm during the truck sequence (or knocked flat by the big mechanic during the airplane fight in the previous encounter, right before the propeller blades finish the guy off), or that scene in The Fugitive where Dr. Kimball stitches up the cut in his side. It woulda gotten to him if he hadn't treated it, but after a little first-aid, he's fine.
Note by my two Raiders of the Lost Ark examples that I tend to think that Indy recovered all his hit points between the flying wing fight sequence and when he takes off after the truck on horseback. I might also claim he used an action surge to recover a bunch of hit points right before doing the slide under the truck, but I don't think that's RAW possible for a character in 5e. But it probably should be.