I like the way you think. I don't think these rules have to be as faithful to the armor rules, though. What do we gain by retaining, say, the three-way light/medium/heavy distinction, or the same don/doff action economy? And as others have noted, how do guns play into this?
Here's a little brainstorm:
Against melee attacks, your parry AC is 10 + your Strength modifier. Against ranged attacks, your dodge AC is 10 + your Dexterity modifier. You can't parry gunfire, but neither can you effectively dodge around when somebody is waving a weapon in your face.
If a class would provide light armor proficiency, it grants +1 to both ACs instead. Medium proficiency becomes +2, heavy becomes +3. These values can possibly be higher; that would result in dodge or parry ACs getting higher than 18, but the other value would still probably be reasonable to hit. Which would encourage trying to attack enemies in their weak style.
"Shields" work exactly as you say, no change there.
Finally, cover gives you a flat AC as though it were D&D armor. Say between 15 and 20 depending on the quality of the cover. You're not really dodging around when you're taking cover, but it's still usually a good idea to do so.