Not sure i would go all the way to remove parts, you may create more problems, i would focus on the remaining problems if there is, after a good cleanup.
You have good cleaning products who can go under the parts and do a very good cleaning.
At my job we use liquid flux who goes everywhere, we rince to board fully after in de-ioniozed hot water with Atron SP200 and or Atron AC205, it remove any greassy stuff and flux residues, and with air spray we remove any water excess and put it in an pcb dryer.