IIRC, on the Amigas the CPU was barely involved in the process of reading the mouse events and moving the cursor sprite on the screen.

Yep, hardware sprites.

Modern GPUs still have hardware sprites for the cursor, and OSes still use them! However, the path between the mouse and updating that sprite has gotten more complex, alas.

I imagine a modern day Amiga would run all mouse and window server code on the GPU, only telling the CPU when the app needs to update its bitmap.