> The poor performer gets singled out, which is just fine.

he's not a poor performer, he's just an asshole. And you can't fire someone for being an asshole

> 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).

You're missing the point. You can't single the person out for violating a policy that you didn't have written down. The only reason that policy is now written down is because that person violated the policy. Singling out someone for being (genuinely) ill is likely to end up with you on the wrong end of an employment tribunal who will ask you "what is your sickness policy"