Thinking a little more about this, the Chair may ask that votes be recorded to show who voted for and against. If councillors abstain to avoid having their vote recorded, the motion is likely to fail. If there is an equality of votes, the Chair may use their casting vote to defeat the motion. My smaller parishes have only seven councillors and we very rarely see all seven together at a meeting. If one person is absent, the Chair only needs two friends to become an immovable object.
A wily Chair losing a confidence vote might simply say "I'm not the Messiah, I'm a very naughty boy* but I won't do it again" as justification for continuing in the role.
(*I use the masculine intentionlly here, as it almost invariably is!)