Yeah it's such a delicate balance to strike haha. The jank is part of the fun, but getting the jank to still feel good is the big challenge. No 2D games immediately spring to mind (i think a lot of them were older neopets and newgrounds games that arent currently available) though I'm sure at some point while I'm in the middle of a totally unrelated task I'll randomly remember some older ones I used to play on the web that had the good goofy balance of jank and reward for overcoming the jank, but I did think of how some 3D physics based games have been handling something similar.
Of this jam, Nuclear Mayhem has the "jank as a feature" and games like it. The struggle to control things properly because the weights and collisions are off just enough that you can't beeline it from point A to B but it's still doable, while antics ensue in the middle. I think in their comments, some people mentioned Fall Guys, which has a similar thing going for it.
Something I think could elevate the absurdity is possibly character reactions to the failures, like maybe an icon of the character in one of the corners that looks just as perplexed as the player probably did that something failed in the way it did, but also provides slight validation to the player when they overcome it. There's loads of different things you could do tbh.
As far as the mop goes, I do think it was specifically Unity handling mouse input a little too overzealously because the web build was SO ready to chuck the mop the second you got any momentum at all. I think you guys are on the right path and with some testing and time you'll get the wonky controls to that "just right" state so personally I'm not worried about it at all. If it ends up continuing to be an issue while you're working on it, I think having a setting somewhere to control mouse sensitivity might help.
While I was typing this I was thinking about how poorly Joust and Joust 2 played but how they are probably some of my all time favorite arcade games because even though they control like the hottest garbage on the planet, they are so fun to play thanks to the little things like killing an enemy or even your friend with your ass because they ran into you too fast while you were turned away.