I can't really prove I used the ability before Puppet died but it's definitely a replicable bug
Edit: I played a few rounds and this bug seems to only apply to the Puppet, the rest doesn't let you use their abilities if they die
The edit doesn't sound true, unless you mean being able to use a ability after the "Village is safe!" text appears.
Otherwise you can always use a usable ability even if the card is dead.
I just played 2 villages and instantly stabbed a Fortune Teller, and in another a Jester, and both allowed me to use it's ability.
In fact I don't ever recall a time when a dead card refused to be used if it has a usable ability.
So it's not a bug at all, nor a glitch, it's part of the game.