If we really want classes to have a diverse set of first-buy items, let's give every champion their own unique item (142 because Viktor already has his). The items could be adjusted on an individual basis, just like the champions, and could be given ability-specific effects akin to Talents in HoTS.
The issue that would arise, though, would be in making each individual item better options than the existing first-buy items. Though that could be done by cutting the stats of all the existing first-buy items, so they don't just give everything the champ wants. Cut CDR off of most of them. Cut the number (or effectiveness) of unique passives, replacing those with the ability-specific passives of the character-specific item.
Would this be an insanely huge update and start as a balancing nightmare, basically changing the game on a level that it's never been changed before? Hell yeah. Would it be cool? I dunno. Would it make builds look a little more diverse? Yeah.
Give Lux her wand, Akali her scythe, everyone their unique swords, Brand his... Cinder? Like Bami's Cinder?
As a little bonus, it'd give names to the weapons of all the characters whose weapons don't have names. Or it could be a joke, like just naming Yorick's "Shovel".
Edit: If Runes Reforged showed us anything, Riot's not against changing every single champion in such a way. Remember, they did have to individually adjust every champ's base stats to account for the loss of stat runes, on top of the yearlong scramble to adjust the new runes.