Technically, your death was a certainty until the host decided to manufacture tension.
The thing is, is that there's nothing for me to gain by fakeclaiming investigator in this case. The mafia only get the extra member bonus if they kill the cop with their night action, not through the vote. So if I was scum and making this play, I would have to know with near-complete certainty who the investigator is so we could kill them that night. However, that's assuming that individual is indeed the investigator and will not have doctor protection.
Either way, it would just be a trade-off and the mafia would gain no numerical advantage; the town would automatically lynch me the following day, so the extra mafia member would only be a replacement. At best it's just a stalling technique, which isn't really necessitated this early on in the game.
Coming forward as investigator to lynch you off can bring about a townie alliance; if I, as the cop, am in the majority vote against you, I gain the names of two confirmed town members. If the doctor outs himself to me through PM, that's a solid block of four townspeople right there, all validated. Given that the last member, the Godfather, has investigative immunity, this can really help weed him out over time and make this game much easier to finish.