I think the D&D Multiverse is unique from the "Real World" in that most species were literally created by Powerful Entities (often "Gods") which are a known thing. We're not talking about millions of years of evolution here. We're not talking modern Real World taxonomical science with genus and species and the like. The Powers that Be made original choices as to the special abilities of creatures they were creating or modifying, and perhaps later they (or other entities) made some further modifications to those species, making offshoots. "Oh, these elves need to live underwater? Your tribe of elves is blessed, and now you breathe water." Boom. Done.
High Elves are High Elves because they are created/altered to be a specific way. Same goes for Wood Elves. And Tieflings. And Dragonborn. It's a fantasy world where shapechanging and reincarnation are real, quantifable effects, and different species can interbreed. It would be nice if we could get over the real world assumptions and correlations of "race" and "species". Somehow accept the fantasy. Get the real world turkey jerky out of the fantasy peanut butter. Gamers still want options. We're just a hung jury over how to name those options.
The game's history has built an entire foundation on the concept of "Race", and variances within a race (called "subraces") that gave different racial abilities and ability score modifiers, mainly because gamers like options for how their characters are depicted. Tolkein and many other fantasy authors, as well as game designers, made race and racism extremely important in their worlds (and sure, their stories are often about disparate races working together). Just look at how half-orcs and half-elves were introduced in D&D. It wasn't enlightened design. The early designers basing their game on these themes didn't know how problematic it would become, and how the meanings of words evolved, and how the world itself would change how they would perceive those words.
Because this is a game played by real world humans, we still have to deal with the naming conventions that originate from those negative real world origins. And that is what makes it hard. We've gone through most of the real world words. They all have connotations that can mean other real world things, and mean different things to different people. We also have people who are fine with old naming conventions even if they are problematic to others because it doesn't matter to them if it is a problem for other people. They like what they like.
As for a solution? There isn't an easy one. Wizards will have to make a choice, and people will accept them, or hold a grudge because their personal perspective wasn't what was chosen.
"Race" doesn't bother me as much because it has been natural for the game and my lexicon for so long, but I understand that it bothers others. I think "Species" is a bit better, but only if we can break it loose from the real world taxonomic definition into a more colloquial use. But colloquialisms don't occur overnight. If one of the definitions needs to become more prominent and accepted, they need to start somewhere and use it in the game so the new generations use it. As stewards of the game, Wizards has to make decisions that will impact the game 50 years further down the line after people like me are long gone.
When it comes to the Half-species debate, as a general term, I find "Mixed" to be more inclusive terminology over "Half". "Half" triggers folk in multiple ways that "Mixed" doesn't (including the derogatory connotation of others calling a person half-breed/species; as well as Half meaning 50% and almost no one is exactly 50% anything).
When it comes to a group of mixed offspring/family, I also like "Kin" because it sounds more familial, welcoming, and openly inclusive, and it is not necessarily limited to a specific "Species". For instance, I like "Elf-kin" over "Half-elves." I think Elf-kin can be an entry in the Elf section.
Maybe add the Elf-kin to the Elf, and move the contentious Mixed Race rules out of the PH, and plan to create an optional Species Encyclopedia book that can dedicate an appropriate amount of space to the matter, rather than a Reminder box in the PH. The Mixed Species rules don't have to change the Base Species of the PH. They can add alternative ways to see the Species, like the 2014 Variant Human was still a human, and the Legacy content Species are still options.