Frankly, I think the current approach is just hard-baked into D&D culture since 3E. It actually is similar to what you describe -- whatever the 5E PHB says, I very seldom see anyone at the table say "Can I make a Charisma check with Persuasion?", they always say "Can I use Persuasion?" And yeah, flexing the required ability score is cool, but there's a huge obstacle to running the game that way: the character sheet. Your character sheet says "Persuasion +7", and this is a combination of your Charisma score and your proficiency bonus (or skill ranks, or whatever), so you can just grab the number and add it to your d20 without thinking about it. Making an Intelligence + Persuasion check requires extra math. Even if you redo the character sheet layout so the "Persuasion" entry is score-agnostic, it still would require extra math. And yeah, to some of us, the thought of grabbing two numbers and adding them together might not exactly be daunting, but other players do struggle with keeping track of that stuff and would rather just take the number and roll the die.
So a lot of the time what I do in circumstances like the librarian conversation is leave the characters' numbers alone but make the Persuasion DC 10 for the wizard and 15 for the bard.