The flat hand spring is the #1 repair part needed. Being a bit on the thrifty side, I've been known to refit a broken spring and keep getting service until it really goes to pot. A broken hand spring won't prevent function, providing you keep the muzzle lowered while cocking - fine until you finish a match/shooting session, so I don't fret over a failure. For spares, I've bought assorted parts from Dixie Gun Works and other sources over the years. No problem at all to use Colt hand springs from C&B 1851-1860 models (re-form as needed), or trimmed-down cartridge hand springs for the Remington. In fact I have a box filled with partially completed Colt-style hands (another do-it-your-self project) and find they can be shaped by grinding to nicely fit the Remington in a pinch. (A coincidence that Remington's design used a sub component of the Colt gun-works?).
Come to mind, I do tweak the sharp bend on the Remington hand spring into more of a round. Seems too sharp to me, suppose I remodel to look more like a Colt...but then, I've been know to substitute a Colt part anyway. Also, look int the hand channel to see if there is casting debris or sharp edges.
As far as mainsprings, I prefer to reshape/regrind existing mainsprings before dropping $$ on a more expensive "lighter" mainspring. The trick is knowing not to overheat (too hot to hold = dunk in cold water), & when to stop... as too-light just ain't gonna work. If this job is botched, a new replacement can be placed with the next order for spares. Remington is easier than Colt due to the adjustment screw.
Many have a hard time believing I own cartridge guns....but I do...honest
Slim