Let's look at this from the other side of the game.
Imagine if as a spellcaster, you get to pick your cantrips, but every leveled spell you get is randomly determined per day. Not in the old "magic user randomly generated their spellbook" style, but more a "here is a deck with all the first level spells in the game. Draw four. Those are the spells you have until you finish a long rest." Maybe you get grease, burning hands, cure wounds and shield today and tomorrow you are stuck with detect magic, longstrider, animal friendship, and heroism. Make it work.
1. Would you want to build a caster who you have little or no control what your abilities are day to day? You might get useful spells or you might get useless ones?
2. What if you were interested in a theme (fire mage, healer, or like) and ended up with no spells dedicated to your theme?
3. How do you plan strategy or tactics if you cannot know what your character is capable of day to day or round to round?
To me, round robin weapons would be a lot like randomized spells. It might be fun if the goal is improvisation and quick thinking, but like how not everyone wants every meal to be an episode of Chopped, not everyone wants to be constantly switching from archer to shock trooper to duelist round to round.