"Martial Arts", hmmm, let me spoil your party (and be the smartass).
Using weapons is "Martial Arts", and far more effective than unarmed strikes.
Believe me, I tried it (only in training!).
Gimme a good kitchen knife and I could probably kill a real world karate champion.
Anyway, real-world-smartassery aside, I think the tavern brawler feat does exactly the right thing.
The above mentioned feat is also okay, except for the finesse thing.
Dexterity to damage is a big mistake of 5e IMO.
Again something I learned from my martial arts (weaponless) training:
If I had to go to a (not only fist-) fight and I could take some guys from my gym, I would take the heavyweight guys with me, not the lightweights. Mass hits harder. Period.
Oops, even more smartassery...