Ahh. This is where the issue lies.
To me, mechanics don't ever tell any story. There's no process involved and never has been, AFAIC. The races were given a bunch of fairly arbitrary abilities that kinda/sorta said "this" or "that" race. So, no, the mechanics to me have never mattered as far as picking a race goes. Pick whatever gets you the results that are close enough and move on.
So, I look at the fact that a 2014 half elf is virtually indistinguishable mechanically from an elf and think, ok, just make them elves. Poof. Done. Or, if you want to go the more Tanis Half-Elven route, pick human and be done.
It's up to the player to portray that character as being that race. Whether I've got a +2 to stealth does not define a race.
There was no process before because everything they designed to be playable was given its own bespoke mechanics.
Want to play an elf? Pick an elf.
Want to play a dwarf? Pick a dwarf.
Want to play a Khoravar? Pick a "half-elf".
Want to play a lupinal? Er, we don't have "monster PC" mechanics in 5e, but talk to your DM and see if the two of you can homebrew up a solution.
"Pick-a-parent" introduces a process where one previously didn't exist, applying specifically to characters of mixed ancestry. It does so for a perfectly noble reason, because creating bespoke mechanics for every possible combination of mixed PC ancestry is already practically impossible and only grows more so with each new addition to the roster (combinatorics be like that), but it means that now, when you want to play a mixed ancestry character, the game no longer directly gives you your species mechanics and instead tells you how to derive them.
It does this in the simplest possible way: pick one of your ancestors - that's what you are now.
That's where the process and its story and all the implications that spring from it come in. Want to play a lupinal? Well, we still don't have playable "monsters", so you and your DM will have to homebrew up something bespoke. Want to play a Khoravar? You're an elf now - get over it.
Obviously, a character's species mechanics are in no way the sum total of what defines them. I am not claiming that you can't roleplay a "half-elf" while using full elf or human species mechanics, and that may work just fine for a lot of people.
But it causes problems for me. It creates a disconnect between mechanics and worldbuilding for mixed ancestry characters that doesn't exist for single ancestry ones, and forces the expression of mixed heritage entirely into the roleplaying side of the game.
If I wanted my character's ancestry to have no meaningful impact on their mechanical identity, I'd use a Tasha-style Custom Lineage for everything or just play a game system without species mechanics at all.