Well, yes, ... and no.
I fully agree that as soon as the fighter gets +3 armor with a +3 shield, the game goes out of whack.
But a simple limit of "max three things with attunement" just becomes so restrictive it ceases to be any fun. When you already have three good attunement items, every new piece of loot becomes one out of two things:
- vendor loot (because it doesn't represent enough of an upgrade). Even if it is a seriously cool item with interesting powers, it's a hard sell to give up something with a basic bonus to "everything" for it.
- OP stuff (because in order to entice you, new items need to be ever-more "better")
If the attunement system instead had some measure of nuance, it would be much easier to remain enthusiastic about new loot.
Just a quick and dirty example to show what I mean:
Imagine you have 10 attunement points.
Most "major" items that today require (and deserve) an attunement slot now costs 3 attunement points. But now you have design space for items that require only 2 or 1 attunement points.
If the item provides a colorful and appropriate bonus to some secondary ability or enhances your roleplaying potential, but does not make you significantly better in combat, say, that item could be a 1-point item.
Now one player could opt for three "big" items, while another player could go for many smaller items that maybe are "cooler" but doesn't provide core bonuses (to attacks, spell saves, or AC).
+3 plate mail should in this system easily be 6 or more attunement points, driving home the point that you need to make some serious compromises to be allowed such a disruptive item. And if you have even stronger opinions about this item, Horwath, make it an 11 point item...