So, honest question: What about the player that wants to combine different mechanical choices with different narrative choices that are not allowed by the game - because there is supposed to be a trade-off?
Great question, and thanks for engaging and not just arguing.
So, for example, what if a player wants to be an Elf with Stonecunning instead of Keen Senses, or a Halfling with Relentless Endurance instead of Lucky?
I see two differences between those and the ASI question:
1) In my opinion, those racial abilities have distinctive racial connotations. +2 Dex is a specific mechanical benefit of Elves, but it's not uniquely flavorful (to me). Not only do several races get an identical benefit, but all races have a Dexterity score, and all of them can get a 16 Dex by 4th level if they so choose. So, no, a 16 Dexterity just doesn't evoke Elvishness (again, to me; YMMV)
2) But more importantly...and less subjectively...the mechanical value of +2 in your primary stat puts too heavy of a finger on the scale in terms of race:class synergy. If you're weighing Halfling Luck vs. Relentless Endurance you can rationalize as better whichever you really prefer. And, yeah, you might just have to decide whether you really want a Halfling, or if you really want Relentless Endurance. Either way, both are quite useful for rogues. But if you're making a Rogue and weighing +2 Dex vs. +2 Strength there's not really any question.
Now, if we noticed over time that we were seeing an improbable number of half-orc rogues (after the initial surge caused by the novelty of it) we might need to reassess how powerful Relentless Endurance is.
So I think it comes down to a kind of cost/benefit analysis: how much flavor are you getting (in terms of abilities that are reserved for certain races), at what kind of cost (measured as the incentive to choose particular race:class combinations). And in my analysis (and apparently WotC's) fixed ASIs provide less flavor than abilities, but create a high incentive to choose specific race:class combinations.