Rob;
No offense, my brother, but you clearly need to find out a bit more about what you're doing with this gun. You've made a very good choice in coming to this forum for that info.
To start, the firing pin is SUPPOSED to make an indent in the PRIMER which is located at the base of the CARTRIDGE. The BULLET is just the part that is propelled from the barrel when the cartridge is fired. The Model 627 in .38 Super does indeed use the .38 Super Auto cartridge, not the .38 Super Police. I'm not fully knowledgeable about the M627 .38 Super, but it may be that your revolver REQUIRES the use of moonclips to fire the cartridge properly. That may be why you're getting light strikes, not enough to ignite the primer (although you didn't say anything about light strikes).
The .38 Super is a semi-rimmed cartridge, which is why it can work in a revolver, though there are other guns set up for fully rimless rounds.
I think you may have bought a highly-specialized revolver, and need to get together with a real expert in that model to use it to best advantage.
Welcome to the forum, and please do keep us informed of your progress.
Larry