Replace our custom drag&drop code with egui's API for it

[Adanos020]

Is your feature request related to a problem? Please describe.
Having this feature implemented manually comes with a lot of problems. Not only does it make the whole codebase a lot more complicated, but it's also prone to breaking changes from egui. One such breaking change was introduced in egui 0.26 where the user interaction logic was rewritten, leading to recurring bugs like #226 that are hard to track down.

Describe the solution you'd like
Ideally, have all of our d&d logic removed and use the API provided by egui instead. Realistically, some bits and pieces would have to remain for proper handling of tab drops, but all of the input handling should be done by egui.