I mean a functional, noncosmetic, advantage. If I left the rust that was already there, would it weaken the parts? If it would cause more damage to some parts, which ones?
The effects of rust are typically overstated. Where is you concern? Remove rust (or replace the parts) on moving components such as chains and cables, they will run smoother, more efficiently and quieter.
On non-moving surfaces such as frame and handle bars etc, forget about it. Under normal use high quality steel gets nothing more than cosmetic rusting, and low quality steel bikes are so over engineered the loss of strength due rust won't make any difference to the serviceability of the bike.
If you ride the bike in winter where the salt roads, or at the beach, then you may want or need a bit of protection, however the moving parts will be rusted solid a long time before the fame fails.....
Another case you may want to consider is if the risk of rust stains on clothes and such like is a concern. In this case then treatment might be needed.