I wouldn't mind that either. But that's not how the game is really built. Most of the classes that get good "I don't take damage at this time" abilities are second-line or even back-line types (Rogue, arcane casters with Shield or Silvery Barbs). You have to literally spec into it as a Fighter, either by taking the cruddy +PB to AC Dueling Feat, or becoming a Battlemaster and throwing one of your mastery dice at an attack.
And if players do go out of their way to get things like shield, I see a ton of DM's gripe about it. I think it's a psychological thing- we'd rather do a boatload of damage to a raging Barbarian, despite the fact he's basically ignoring half of it, than attack a player and have them turn a hit into a miss, lol.
Honestly maybe the solution is to cut monster damage or increase player hit points if you want to shed in-combat healing. Unfortunately, having Clerics heal people is so ingrained into the D&D experience that we're stuck on this swinging pendulum between "healing too good" and "healing not good enough", because the developers can't seem find the right balance.
I'd prefer a design that just says "you are assumed to start every fight at full hit points, but sources of in-combat healing are exceedingly scarce", but that path was soundly rejected, so here we are.