I like the idea for arcane spellcasters, but most divine spellcasters automatically have the whole range of clerical domain spells right at their fingertips. So it wouldn't be consistent if one type of spellcaster started off not knowing their spells except through exploration, and than have different rules for the divine casters. We could solve that by eliminating clerics, druids, paladins and the like, and making it so that only spellbook type divine casters like the Archivist, but that would be too limiting.
I have a twist on your idea, however. All characters should have skill points in Use Magic Device so that they can use wands and so forth. Arcane spellcasters won't find many books and scrolls since a lot of knowledge has been destroyed. However, they can learn a spell from a wand, rod, or staff if they use the spell effect a few times and then reverse engineer the magical spell in order to put it in their spellbook.