CubicsRube
Hero
During the 5e playtest, races gave +1 to two different ability scores (usually +1 from the core race and +1 to something else from the subraces) and classes gave +1 to their primary score. So all Fighters got +1 Strength, all Wizards got +1 Intelligence, etc. This worked excellently throughout the playtest process and I have no idea why they changed it in the published version. This also seems very similar to your suggested rule. I see no reason your rule wouldn’t work just as well.
I like this rule. Didn't know about it.
It makes a lot of sense to me that someone trained as a fgihter would be stronger or quicker, trained as a wizard more intelligent. As a bard, more charismatic, etc