Another good grapple build is a Barb/Rogue MC.
The difference between wear and wield is the sort of thing that only matters for builds on paper, because if you try to bring that logic to your table, you're just going to get laughed at.
As for building a grappler, a shield is actually just about the best thing you can have, strictly because of Shield Master's bonus action shove. Grappler as a feat is highly disappointing, and Tavern Brawler is similarly a bit lackluster. And the two feats don't play well together, since both Tavern Brawler's grapple and Shield Master's shove require the bonus action.
Also worth remembering that this isn't 3.5, and feat slots are extremely fucking pricey...
Having both Shield Master and Tavern Brawler gives you more options, which isn't a bad thing. Or you can skip Shield Master entirely and just go Battlemaster with Trip Attack. Although if I played Captain Faerun, I'd pick it up later for the other two benefits of the feat, which are the real reason to get it. As for feats being pricey, that's another reason to go Fighter and/or Variant Human.
Yeah, its pretty damn bad Munchkinry to say wielding a shield is not using it as armor. It specifically says "wielding a shield increases your Armor Class by 2."
Persus, what benefit would Captain Faerun get from the Tavern Brawler feat? Keep in mind that it only allows a bonus action grapple on a 1d4+Str damage hit. I'm pretty sure if you're going for a grappler, there's better things to do than wear a shield.
And, yeah, I agree that actually making him a class that's supposed to be strong would be best, and those have shield proficiency anyway.
Tavern Brawler is mainly for proficiency with improvised weapons. And since anyone calling themselves Captain Faerun should be using their shield as a frisbee/improvised weapon, that's pretty useful to have unless your DM allows your shield to be considered something else.
Damage at that point is just icing, they're already taken out of the fight.
Until you start fighting spellcasters, that is.