As an un-helpful aside, I don't see what'd've been wrong with all permanent items using attunement.
I think the problem with that is that it punishes players for wanting to use "fun" items. If you're a Fighter and there's a magic sword, magic armor, and magic shield, you're kind of obligated to take those over fun things like a horn of blasting, want of missiles, etc. even if the martial equipment is only +1 do-nothing. Worse, at some point, you won't take any rewards unless they're essentially direct upgrades. That's just poor reward design and kind of subverts everything fun about the weird magic items.
And as much as it makes sense to say, "Well don't have +x do-nothings then," they're kind of built in to the lore of the game system at this point. Armors that are magically better at being armor, and swords that are magically better at being swords are really what people imagine when you say "magic sword" or "magic armor." That doesn't make them fun or interesting to use.
The longer I play 5e, however, the longer I'm favoring going back to the paper doll approach, perhaps just with fewer slots or a higher limit. I like the idea behind attunement, but, it's not very fun the way it's implemented. Kind of like concentration,
everything interesting has attunement. It doesn't matter if an item is so niche that might as well be a potion or a scroll. It makes items like
trident of fish command, a +0 trident that has 3 renewable charges of
dominate beast that only work on creatures with a swim speed, compete directly with a
sun blade or
ring of protection or even a
+1 trident.
We've played a few dungeon crawl games of 5e with rolling for items and treasure tables. On more than one occasion before 10th level we've found magic items that were fun or interesting or useful in corner situations, and nobody could use them. Everyone had items that were more powerful or that offered too much benefit to give up. It feels really stupid to roll magic items and get results that are cool, interesting and fun, and then
still not use them.
Say:
2-3 weapons/rods/staves/wands/shields/misc held items/rings
2-3 clothing/body/armor/misc worn items/rings
Another option would be to say that the number of items you can attune equals your proficiency bonus. Honestly, this is probably just the easiest solution if you want more items in your games.