Multi classing is optional. Just don't allow it and the problem goes away. Multi-classing is very 3e/4e(sort of..)/5e, but not very Birthrighty.

I was envisioning Eldritch Knight as an elf/half only class, and perhaps let that subclass get 1/2 from sources or even full rp.

You can do something similar for arcane trickster as representing the magic-user/thief combo of days long ago.