I don't think there's any logic to the save system to begin with. Why do wizards have good Will Saves and Fighters poor Will Saves? I mean, if you name a novel where a warrior protagonist stands up against evil sorcerors, that warrior inevitably has an indomitable will. In contrast all kinds of mythological wizards get mind controlled by the djinns or demons they summon.

I don't think there's any balance or flavor reason why characters have to get the saves the books give them. You could just let everyone pick two strong saves at level 1, and keep those regardless of multiclassing. Let Monk keep three, as long as Monk stays his highest class. Give the bards and rangers a couple free hp. Give the cleric one extra silver piece.