Several people have responded why WotC changed their initial stance on unarmed strikes several years ago, so I am quite content with the reasoning for the change (even if, personally, I don't agree with it).
I think the revision is very limiting in the spirit of the game, imagination, flavor, and fun. While certain issues
would arise from having unarmed strikes "weapons" again, I don't think any of those would be insurmountable. All that being said, I also agree unarmed strikes should be limited when facing a foe with a weapon unless the character has more advanced training (monk, tavern brawler, or grappling come to mind).
One of the major advantages the designed gave unarmed strikes in 5E was the fact that no OA is allowed by the target if they wield a weapon (or have natural weaponry in the sense of fangs, claws, etc.). Oddly enough, this is one change I don't like in some respects. It does go against realism. Like Jackie Chan (and my own teacher) said, "if you are facing a person with a knife, run if you can, fight if you must."
In the spirit of more advanced training, I think unarmed strikes by more "normal" people should either be with disadvantage (you are being cautious) or allow the defender an OA if they have a weapon in hand or natural weapon (other than their own unarmed strike). A simple trade-off would be if someone has advanced training, their unarmed strikes are considered weapons in all respects, and therefore can attack normally and do not provoke an OA. If you think in terms of getting advanced training, it will likely cost you a feat or maybe add a fighting style (as suggested by
Salthorae) or taking levels in monk.
Personally, I think the cost would be worth the benefits and allow the flexibility to make a better brawler character. Also, as I think I mentioned once our table has a LOT of humanoid opponents, with only an occasional "monster". So, I am not as worried about punching a lion or kicking an elephant and hurting them much. I wouldn't expect it.
Then again, the more I play D&D, the less I like characters who are "heroes." For the most part, a 1st level fighter has better than a 50/50 chance to kill a lion... Granted, I am assuming the fighter would have a good AC (say 18 from armor, shield, and a bit of DEX) and is a sword and board type. Still, it seems sort of... oh, wrong?