Nice save on the old Peugeot!, if you are going back to an original shift set up this
link might help you see how it goes together, if you have a local bike co-op that would be a good source for that type of vintage part, although those Suntour stem shifters work very well and imho are as good or better than the simplex, although certainly not stock. And be
very careful with the front derailleur, they have a derlin plastic body in that era and are prone to cracking if you look at them wrong.