How would that house rule impact the choice you make?
It would open it up somewhat, but it wouldn't be night and day, for me.
There are a few races in D&D so bad that I wouldn't even consider them, especially not after free stat choice. PHB Dragonborn are one. But there aren't many. Maybe a double-handful at most.
And I tend to like outre races, and in 5E, those have also tended to be pretty good and powerful. Like, I might well play a Satyr or Yuan-Ti based on their basic concept alone - indeed, the Satyr illustration (I know it was stolen from MtG!) is so fantastic it made me immediately want to play one!
This one:
Full of energy, joy, life, fun.
Not this tired, generic and forgettable nonsense:
YAAAAAWN! Please no more "floating on a white background" race illustrations EVER. PLEASE WOTC. I BEG THEE!
You know what I'd never play again without it offering a mechanical benefit though? A BLOODY ELF. Half-Elf or similar, sure. Not another bloody elf though!
But you theory is, I think, broadly correct - most people, most of the time, are picking races primarily for the vibes, for the roleplaying, for the cool ears, and so on. There are certain races that are elevated or pushed down by being mechanically awful or great. Elves in 5E, for example, are simply boosted wildly by being overpowered. As they have been in most editions. Take aware the mechanical advantages, including their sleep advantage, and they'd still have devotees because they're pretty, anti-hirsute, extremely long-lived and so on - but I think you'd see a significant drop.
One question is would we see more or fewer humans? I think we'd actually see slightly fewer, overall, because I think there are a certain percentage of people who mostly pick human out of fear of making a "bad choice" race-wise or because they just don't want to get involved in the analysis of picking the "right race" for a class or whatever, and humans are fine. But if every race was the right race for a class or setup, then I think things would broaden out a bit.
Also realistically media trends are going to vary stuff - if we see D&D races/species as "stocks" as it were, for Dwarves are going to be way, way up right now from their usual value thanks to Delicious in Dungeon and the beloved Senshi and the badass Namari. Hell even Halflings will be going up thanks primarily to Chilchuck.
and that includes for the story telling, as having mechanics that reflect the narrative of the story increases the validity of the story
I think a good case could be made that 5E doesn't really do that very well or consistently with regards to race. In fact, major story stuff about races is often just not really reflected in the abilities of a race, or only lightly. Other stuff which is barely a part of the story of the race, or just happened to be a mechanic in a previous edition is often given top billing. We might look at Yuan-Ti Purebloods (now just Yuan-Ti, I see) for example - magic resistance is not a magic story theme or even really something people are naturally going to think of re: them, but it's their most major ability.