Description of the feature

When rightclicking with a shulkerbox on a storage drawer, it should take out all the items of the shulkerbox, which match the clicked one. When clicking the storage interface, it should automatically sort all items into the system, just as clicking twice with an empty hand would.
Optional addition: When retrieving items with a shulkerbox in hand, they should automatically be added to the shulkerbox, allowing to fill the entire shulkerbox at once (max of 27 stacks).

How does it make the mod better?

This would allow to bulk store and retrieve items without having to place down shulkerboxes, drop the items, and manually inserting them. Yes, shulkerbox unloaders do exist, however, you'd need filtering etc and this would make the QOL way better.

I see one major issue with this idea: How do you differentiate between a player inserting a shulker vs them inserting its contents? While you can't by default store shulkers in drawers, that option can be enabled in the config.

Edit: Just woke up, sorry, overread the part with the config :D Added a new idea below

Maybe a sneak modifier could be used?
If a player sneak right clicks, it dumps all the contents. Certainly, that would disallow placing shulkerboxes against the blocks. But that would be the lesser evil since it could be disabled / enabled via an config option.
Or a new block is added, which only takes in shulkerbox right clicks, as in a new "bulk" storage interface for insertion / withdrawal.

The config had some face interaction behavior config iirc, maybe it could be included there.

Another question - where can shulkerbox insertion be enabled? I didn't find it in the config for 1.20.4 fabric

Shulker box inserting isn't an option in and of itself. It's included in the option to allow nesting storages. There's a big warning on it in the config screen about data sizes, but it should work fine unless you do stupid things