Really, unless you just want to boost scores, why bother adding anything for any of the above reasons?
For many players, IME, seeing a logical increase due to the make-up of a race makes sense. Likewise, if your character has a background in Soldier, for instance, it would make sense to also gain in one ability that you used/trained with more, such as a physical score. And, of course finally, it makes since that whatever experience you went through to acquire your class would likely lead to a related score boost as well by the time you became level 1. After all, you begin level 1 with some class features that you got from someplace or somehow.
With 9 races, 13 backgrounds, and 12 classes in the PHB alone, you would have over 1400 possible combinations for your character (if I didn't miss any LOL). Obviously there would be some overlap since you have, at best, 216 combinations for 3 ASIs (assuming you allow the same one to be selected up to three times-- such as a getting a Str +1 from race, background, and class...) but that would be understandable.
In the end, it is whatever works best for you and your group I suppose. Cheers.