Unfortunately I just find Rigoletto's plot too depressing to manage. But, I am also into the daddy issues. Yes, it's so terribly cliched, but didn't Verdi do it before it was a cliche?
Miller is the rare example of a Verdi dad who doesn't really have any problems (except those caused by others). If you forced me at gunpoint to pick a Verdi character to be my dad, I'd probably pick him too.
Who is Verdi's worst father? I'd have to go with Montforte from I Vespri Siciliani, whose love for his son is genuine (as genuine as this guy can feel about anything, which is probably not much) yet deeply stunted and immature.
King Phillip from Don Carlo is kind of a dick, but we do at least have the privilege of getting into his head. He is caught in the middle of everything too, which is one of the aspects that makes Don Carlo unique - that there's this other layer of "big bad daddy figure" even above him (the Grand Inquisitor).
And then there's Papa Germont... who, in my opinion, is often not carefully cast. It takes more than just any old baritone to properly play him. It ought to be one of those roles where not every singer is a good fit (like Scarpia or Falstaff), but usually it seems to me that as long as the guy can wear old age makeup and a wig, they'll take him. It needs more than just a proper old man - Germont needs to have charisma and persuasiveness or else the whole opera just doesn't work.