As several vanilla blocks and items can be cleaned with a cauldron (leather armor, banners, shulker boxes), I think a registry for this behavior would be interesting.

The syntax would be something like:
CauldronInteractionRegistry.getDefaultInstance().add(currentItem, resultItem);

Or for even more customization:
CauldronInteractionRegistry.getDefaultInstance().add(currentItem, resultItem, waterLevelsConsumed);

This is already in Cotton, with provisions for mods adding their own cauldrons which can store items for use in interactions or store fluids other than water.

My experience with a system like this in bulky shulkies for my custom shulker boxes has been to listen to the UseBlockCallback and force a success state if my shulker box is cleaned and drop the water level then.

I personally feel this isn't necessary.

I would agree I think maybe the only part that might be a reasonable addition would be some sort of interface for compatibility of modded cauldrons. Something similar to what Boundarybreaker mentioned in Cotton, but even that seems a little niche.

Out of scope and also this won't be needed in 1.17