Hardness isn’t strength though. Hardness is the ability for one material to scratch/cut through another. Strength is the ability to resist deformation or breaking. Silver isn’t as hard or strong as copper, so it would be even worse as a weapon. Not terrible, you could make a sword out of frozen butter if you really wanted to, but there’s absolutely no reason to use it when steel is available. Like you said, you would have to repair it after pretty much every fight.
Hardness of a projectile doesn’t really affect velocity, assuming you’re not firing gelatin or something like that, it’s only down to weight and the energy input. If we assume equal energy for a lead and silver bullet (apparently silver is only a little bit lighter than lead) then the silver bullet would be slightly faster than the lead one. However, silver is harder/stronger than lead, which means it isn’t going to expand in the wound like lead tends to, which actually means you’re going to do less practical damage, even if it might be slightly better at penetrating armor. It would work, sure, but I don’t think it would be as effective as a lead bullet.
EDIT: A quick google suggests silver bullets have been tried and they might penetrate further at short range, but are inaccurate and slower at long ranges.