So the hinges aren’t the issue? i have used heavy duty Flat brackets to hold wood together (also called mending plates or flat corner braces). they are screwed across two adjacent pieces of wood on a flat plane to reinforce joints, repair spots, or prevent wood from sagging - i have used them really effectively on Ikea furniture that wanted to come apart 😅

That is an irretrievable broken joint. No fixing that. I'd suggest contacting a local cabinet maker to make you something similar, but that isn't landfill quality.

Reassemble the door with adhesive and skin the back with 1/4" plywood.

Kregs Pocket-hole jig. Neat,easy and professional. I've used one for years building furniture.