The only argument I could oppose is that when I look, for instance, at profiles for Rurikids with a LNAB in Cyrillic characters, the older profiles seem to have been created by Russian people. Same thing happened when we had Polish or Hungarian contributors, who tended to create profiles with the names they were used to hearing in their native exotic European country. If we decide to use American transliterations (because as you know, French, German, etc. transliterations are different), we are in effect telling all these people to "get lost", and that notable people from their country and of their heritage (I know this is inaccurate, but his is how they would perceive them) do not belong to their country but must be Americanized instead. I'm exaggerating a little, but I'm sure that is how they feel, and you know why. And I'm perfectly aware these people are a small minority on a site like this.
Yes, I know, this is weak and convoluted, and yes, I do think we should be careful how far we want to take "use their conventions not ours". And yes, I know this is an American site (as a non-American you understand this pretty quickly).
Now I'm making a test. I want to see if the profile comes up in search for first name "Theophanu" if we write the proper first name in Greek and Preferred name transcribed in English. Just a test, but I'd like to be certain that the names are not searchable.