No one is trying to figure out doc, absolutely nothing good would come of that. Trying to figure out vig since they've already used their hit (very well, :bowing

and that would allow the town to cross someone off the list for sure. It would be pretty obvious, at least I think, if a mafia scum challenged the vig who role claims. So we either get an innocent for sure and we can all cross them off the list, or we get a mafia scum that challenges and exposes themselves. The doc can protect the vig from a revenge hit from the mafia.
That's my take on it, does that make sense?