Biracial and mixed race people exist in the real world, albeit not in the "part elf" sense, so while there certainly could be improvements to how they are portrayed and talked about, particularly on the half-orc side, I don't really see any reason why half-elves or half-orcs should necessarily be removed.
"Race" in D&D is more analogous to "species", but for better or worse, half-elves and half-orcs have a lengthy history in the game, as do other race options that were generally treated in past editions as being the result of mixed bloodlines, namely the "planetouched" races (tieflings, aasimar, genasi, etc.), where the assumed origin in 2-3.5e was that one of their mortal ancestors had relations with an outsider of some sort.
Beyond that, some settings (notably Eberron) have allowed half-elves and half-orcs to carve out unique, in-setting cultural identities for themselves distinct from those of their ancestors on both sides, so I don't really see them going anywhere.
Ultimately, I think it just comes down to the fact that they would never be able to manage the sheer number of possible permutations, so they have to draw a line somewhere. If you want a half-tabaxi/half-aarakocra and don't want to homebrew, that's probably what Custom Lineage is for...
EDIT: Meant to quote the
@beancounter post you were quoting as well...