I'd flip this.
Picking features that make you better at fighting with heavy weapons grants you a bonus to strength ability checks.
Ability checks are used when you don't have a feature that governs a situation. They are not used when using a feature. So you don't make an int check to see if you can counterspell if you have a counterspell feature.
What this does mean is that if you have features that focus in a specific area, your ability checks related to that area will have good bonuses.
This provides for verisimilitude - a character who uses a huge two handed weapon and smashes foes around will be able to lift stuff better than a book-reading artificer. Unless, of course, the book-reading artificer takes the "lift stuff better" feature, in which case the artificer wins.
This does mean that bards, by learning charisma-based magic, end up having a charisma bonus. Rogues, by learning sneak-based features, have a dexterity bonus. Your attribute bonuses "fill in the gaps".
Now, we could use 5e's "your attributes are defences", but I think a two-tiered defence model might make sense. In a two-tiered defence model, you have your defence (which may be a function of level) - your AC or the equivalent for magic.
And as a second tier, you could have saving throws. Saving throws are resources you expend, where you roll an attribute check in order to mitigate a defence failure. Possibly they are per-day or per-encounter type resources.
Like, imagine if each of your attributes had a 1/encounter saving throw attached to them. And each saving throw can be used in a variety of situations - you can use a dexterity saving throw to dodge a blow, a constitution saving throw to soak a blow, a charisma saving throw to trick a foe into missing, etc. Here, your bonus determines how reliable a saving throw is, with the idea that if you max out one stat you have an "overkill" bonus; but because you can only use a given saving throw 1/encounter, you are better off with multiple decent saving throws, and not just one insanely high one.
The point of all of this is to use mechanics to inform the story.