Heavy Armor would still exist as a way of getting AC without Dex.
And I get the not wanting a treadmil. But, at least when looking at the CR calculator table in the DMG, monster ACs continue to go up with levels. Without magic items, you're expecting about a 65% hit chance against an equal CR monster at most levels (the monster AC even goes up generally at the same levels that player stats and proficiencies go up).
So your ability to hit monsters doesn't change, without magic items; it even falls behind if you pick up feats instead of ability boosts. But the ability for monsters to hit you continues to grow, unless you use magic items to boost your AC.
Now, this might be because there are more ways to boost your AC than there are ways to boost your attack. It's still weird, and I'm not sure that I like it.
I do appreciate that HP means dodging, but since Dex also means dodging and Dex is tied to AC, it's kind of hard to overlook.
Thanks for everyone's input. Bounding AC makes sense, but player AC is more tightly bounded than any other part of the system; player attack scales, 2 or more of player saves scale, player HP scales (though this doesn't feel like much to me considering how much monster damage scales), player damage scales, player abilities scale ... Without magic items, AC caps out by level 8. It just seems odd.
In order to put in scaling player AC, though, we'd either need to base AC at 8 + modifiers, like save DCs, or we'd need to bring back weapon proficiency bonuses, which would seem odd in the system when they already have proficiency bonuses.
Adding in proficiency to AC would require a lot of changes. I roll with magic items, so I'm pretty sure it will be fine in my games. We're going to start a level 11 game soon, so I'll be able to see how high level fights truly feel.