On the card "went mildly mad", the text says "While this text shows, discard one card every time you play a Modifier (including this one).", would you discard a card when you played a modifier to cover up this card already played on a family member?
I'd say that because you discard when you first play Mildly Mad, you don't discard when you cover it up with a new card.
No. Play the card, then consider triggered effects; since the text of "mildly mad" is covered up at this point, it no longer applies.