Perhaps a silly question but… if you really want to avoid synthetic glues, maybe figure out a way to use screws? I’m not joinery expert but it seems like the best way to avoid synthetics while also being very mechanically stable.
Indeed, however my concern would be spilt liquid getting in between the boards. There’s no where else to eat in my rental other then at my desk, so I basically live there. ༎ຶ‿༎ຶ