If the town lynches the mafia member today, or if the town lynches a townie today, then it does not even matter if the mafia member is the CD... in either case the outcome of the game is certain (either because the mafia will be gone, or because the remaining mafia member will win).
However, if the CD is alive, and the town does nothing, then here's what happens:
1. Town does nothing.
2. CD kills a townie at night, leaving two townies and the CD.
3. Day breaks, and the CD imprisons a townie, leaving a stalemate.
4. Night falls, and the CD kills a townie.
5. Day breaks, and the previously imprisoned townie is the only townie left, leaving a stalemate.
6. Night falls, and the CD kills the last townie. Game over.
So if we operate on the assumption that the CD could be alive, we absolutely must pick the mafia member correctly today... otherwise, we lose either by a bad lynching or by no lynching.
If the CD is dead, then we could lynch nobody today, and still have a chance to win tomorrow.
That's a heavy choice for us. Lynching nobody is a gamble.