How old is the piece? If it is antique it is assembled with hide glue. If so the glue can be reversed by applying heat and moisture and the piece can be disassembled and reassembled easily.
There is no need to remove the old glue as long as you reassemble with hide glue again. I would use hot hide glue where time constraints allowed it and liquid hide glue where I needed assembly time. If it is really an antique this would maintain its value and return it to its original state.

The early bird gets the worm but its the second mouse that gets the cheese.