Class specific lists can yes, but the way they did that successfully without making more casting focused classes feel like their toys were being copied by classes with more class powers was 3.x's class specific spell levels. If for example wotc put out a bovd-lite with necromancers & other evil casters in it there could be warlock & sorcerer archetypes that add any meaningfully thematic spells like the bard's songs of restoration without needing to give all of them to every warlock & every sorcerer even though they might be available to every wizard or every cleric as appropriate for the spell.The way I see it, being able to design a spell available to wizards and warlocks but not sorcerers (for instance) is a feature, not a bug. And I think it's going to be easier to do that on a case by case basis than by keeping track of the implications of each spell school/power source combination.
Five works well in MTG because the game was built from the ground up around five power sources with distinct mechanical and conceptual identities. D&D's spell lists are instead seeking to emulate a lot of ideosyncratic genre history. They could in theory be rebuilt in a more systematic way (perhaps something along the lines of @Yaarel 's proposal), but this would require a willingness to break with a lot of traditions, and I don't think the product would look at all like the Arcane/Divine/Primal lists.
I think there are definitely benefits to having classes with extensive spell repetoires (like Wizards) alongside classes with shorter spell lists but more non-spell abilities (like Bards). But I think that customized lists can accomplish this better than shared lists with school restrictions.