Ignoring Mr. Crawford's elucidation of RAI for a moment, I think it's important to see how this stacks up against the Dual-Wielder feat using two normal weapons. If dual-wielding sword and shield beats dual-wielding two swords, I'd definitely go with the RAI and not allow it. If, on the other hand, it is at least slightly inferior to two weapons, I'd consider it.
So compared to dual-wielding two longswords or rapiers, here is what dual-wielding sword and shield gives you:
+2 AC
-2 Damage
-2 to -6 to attack
If your shield is magical that would add even more to AC. If the off-hand weapon were magical that would add more to damage and attack lost.
It's a tough call to me. AC is pretty darn valuable in 5e. And with magic in the mix, it just gets better and better. At higher levels that proficiency bonus and damage look like they might make it superior, but that only applies to a single attack, whereas AC applies to every attack against you.
I'm probably going with Mr. Crawford on this one. A character can shield bash, but not as part of two weapon fighting, so it would require the Extra Attack feature (just like there is nothing stopping a fighter from using Extra Attack to attack once with a weapon in each hand rather than twice with the same weapon). There isn't really much reason to do that, however, unless you realize something you are fighting is resistant to the damage type of your primary weapon but not to bludgeoning.
Good post. Here are a couple of other things to consider:-
The Tavern Brawler feat takes away the '-2 to -6 to attack'.
If you can choose the shield to be a (weapon) for the purposes of TWF, and also choose the shield to be a (not weapon) for the purposes of the Dueling fighting style (and we know that this is the case since you CAN use a shield with Dueling style), it must be noted that there is no rule limiting how often you can switch designations like this.
Although the shield can only be EITHER a (weapon) OR a (not weapon) at any given instant, it's no problem to make the shield a (not weapon) when you attack with your sword (to get the +2 Dueling bonus) and then in the same round make the shield a (weapon) so that you can use it to execute the bonus action attack generated by your first attack.
If you allow shields to actually
be (weapons) just because there is a rule which allows you to attack with (not weapons), then the results of comparing sword & board versus two swords in TWF with Dual Wielder and Tavern Brawler (and Dueling) is:-
S&B = +2 to +5 AC (even without the +1 to AC from Dual Wielder), d8 + d4 +2 = 9 damage
Twin Swords = +1 to AC, 2d8 = 9 damage.
The same damage, +1 to +4 better AC. This cannot be intended.
The solution is simply to enforce the rules. The improvised weapons rule allows you to attack with a (not weapon) without turning it into a (weapon). When the rules say (weapon), they mean it(!). TWF requires (weapons). Problem solved.
If you want to shield bash, you can do so. But not using TWF.