Think the excuse now is our exotics are copies of real exotics, so probably not

Some are exotic because they are canonically one of a kind. Some because they previously belonged to a legendary guardian or character. Some are, as you say, just wieldable enemy weapons. The only one of these that really makes sense as an exotic is the mythoclast as it is a vex weapon that has been modified to be fired by a human. I think the real answer is probably disappointing. They are exotic for in-game balancing reasons and not (consistently) because they hold any importance in the lore.

The earlier D2 exotics were weaker because that’s what Bungie was trying to do at the time, water the game down. That’s why most weak exotics are from that earlier period. What makes the exotic in game, is typically due to them being made with exotic material which translates to abnormal behavior. It’s not so much that they mean something special to anyone. Suros is made with a limited exotic matter, as to explain how it deals extra damage and heals you. Polaris harness Solar power, it’s a paracausal weapon type. Made by regular people during the Golden Age. Not all exotics are easily explained or make much sense. Sky burners is just a cabal weapon. It’s only exotic in the context of it having a limited for humans Like Strum. Regardless of how it was done, kinda pointless to do so with an old gun. Some exotics were just Bungie being silly. Trying to find a new theme, as opposed to the heavy sci-fi themes of D1.