I use black pepper with carbonara. BUT roughly cracked in a pepper grinder. I add it to the eggs, and also crack on more when the dish is done. But NOT the equivalent of a tablespoon for one person. That would be crazy.
Six minutes IS enough for spaghettini. And a bit less if you plan to throw it in the pan with the sauce before you serve it.
I will NEVER prepare any kind of pasta this way. The idea of "simmering" it is grotesque. Maybe I'm just old fashioned, but...