Poor performers get put on PIPs, right? Did that person's poor performance "ruin it for everyone" and put the rest of the working plebs (the entire company or department or whatever) on PIPs? No, of course not. The poor performer gets singled out, which is just fine.

So instead of punishing everyone for some lying asshole's poor judgment, I propose management puts that lazy jerk on their own SDIP (sick day improvement plan).

EDIT: As an alternative, sure, update the handbook's sick policy while that liar is working for you. Since there's now precedent for handbook updating, should be an easy thing to revert it back to the normal, "no sick day policy" after they leave (by whatever means).