Don't see how it makes Insane Doc more likely. Assassin killing today means his team won't be able to kill next NP.
I think i asked that to the hosts in one of my games, either the one i was Assassin myself or my first game when shinobi in my team was assassin.
Anyway, i was under the assumption that the Assassin had a day kill per phase. Meaning that if the mafia team kills N1, the assassin can still kill D2 because it's a new phase. That was my assumption; but the hosts from that game told me that actually no, they considered that an Assassin kill couldnt occur the very next day of a night kill from the same team. I hope i'm being understandable.
Anyway, that can depend on the hosts and how they interpret the role. But from what i understood, now that we have an Assassin kill this day phase, in my eyes it means that the Assassin's team didn't get to kill last phase.
But now that im writting it, it also means that this team didn't kill at all the first phase... Which makes no sense. So it might really mean it's a matter of point of view from the hosts, maybe they allowed what the hsts didn't allow in my game.