That's the point of all of this. It is allowed. It is 100% allowed.
It's just that being a half-elf does not make you a weaker elf with a couple of different proficiencies. Now, you're mechanically an elf. Or whatever you choose.
IOW, every single half-elf is now a unique mechanical creation that is no longer bound by a single definition that all half-elves are identical.
The bottom line is that mechanics=/=in game fiction. They just don't. You can keep banging this drum forever, but, until such time as you recognize that you lost this argument years ago, it's not going to matter. You want mechanically unique half-elves. Fair enough. I get that. But, again, like warlords and various other earlier edition elements, you simply lost this argument.