John, installing plywood will make the old attic floor far stronger. In fact, we used to use this as a method of repairing sagged and cracked ceilings. Removing the old floor boards and then gluing & screwing the plywood down will, essentially, create T-beams out of each joist. Screws about 8-inches on center. Use Loctite PL-series glue and make sure the old joists are cleaned and rough-sanded for good bond. Dead weight of the plywood will not be a significant factor.
5/8" CDX is fine - there is no prescriptive code for this retrofit application. It's whatever an engineer specifies.
If it's balloon-framed, you want to check the joist end connections at the wall studs. That's going to be the weak point. Are they just nailed into the sides of the wall studs or is there a ledger board of some sort? If just nails, those connections should be beefed up, ideally with metal connectors.
Obviously, the joists need to be checked for cracks and repaired with splices if needed before the plywood goes down.
*I don't play one on TV, but I'm a structural engineer in real life 