A +2 to AC is a Very Big Deal. To state the obvious, it turns a 45% chance to hit into a 35% chance.
If +2AC is a Very Big Deal, what do you think about an attack that gives your target a minimum of +2 and a maximum of +6 to AC?
Because that is what happens if you attack someone with a shield, without having the Tavern Brawler feat to support it.
In my opinion, the penalty for attacking with an improvised weapon is already high enough that it won't happen much.
As for the written rule about +2 just for wearing a shield, note that that rule is written with the supposition that you're not constantly swinging it about, trying to inflict damage with it. Otherwise, it would be listed on the weapon table and be given a damage expression.
You're reading things in to the written rules that don't actually exist, and claiming to know what the designers intended. In this case, they're pretty responsive - why don't we just ask them on twitter?
In the general case, this is just a ruling. I'm hewing towards the RAW in this case because I think the penalty is already enough to discourage bad behaviour. You want to houserule something that you are perceiving to be an issue. I'd say wait and see if anyone actually tries it in your game, but obviously it doesn't actually matter what either of us think; we're going to go off and play the game the way we think is right, and that's perfectly good.
Yes, you can use it to shove and retain the AC bonus, but only as a feat, which is another Very Big Deal. Taking the feat means you've put a lot of extra training into being able to do this particular maneuver without sacrificing defense.
To be able to do the same thing as the existing feat, only deal damage instead of shoving, should require at least as much sacrifice as the shoving maneuver. Which means it should probably require a feat to learn how to do it.
It's not a question of whether it's possible to attack with a shield -- no one doubts that it is possible. It's not even about applying a temporary penalty when you do use it that way. The question is, what does it take to learn how to do such a thing in the first place?
I know that in the real world, I personally could never in a million years deal any real damage with a shield I was trying to use defensively. But I can easily imagine someone training hard and learning to do exactly that. It wouldn't be part of learning to use the shield defensively (i.e., proficiency). It would be something extra -- something like a feat.
This is a perfectly fine, consistent houserule.
But still a houserule, by my reading of the rules. While you wield a shield, you get +2 to AC. There are no conditions. The only limitation is that you can't benefit from more than one shield at a time.
But seriously, this thread has turned into one of those silly places where people with different expectations of the game repeatedly try to convince other people that their interpretation is superior. It's not, for any of us.
I don't believe there's any chance I'll convince you, Hendrix or anyone else who seems to have a visceral "Nope" reaction to the RAW 5e shield that I'm right, and frankly I shouldn't even try.
I'm mildly curious enough to ask the question on Twitter though, so I'll do that. And, obviously, I'll ignore any such official ruling if I don't agree with it. You should, too
(Although I
will accept that my preferred version is a houserule if there's a ruling that goes against it. I am magnanimous like that)