If the beat-up saddle is comfortable, keep it.
Fit some 1.5" slick tyres, new cables. Service the hubs. The bottom bracket would be a Shimano cartridge, so if it is fine, leave alone.
Brake upgrade is worthwhile, cantis are tricky to get right, but if you are happy with the setup, keep.
I upgraded my flat bar to an On-One Mary with more rear sweep.