Yeah, it's an unreasonable request. I don't even know how to type that symbol. It isn't on my keyboard. Sure, I could probably find a way to type it or copy it from somewhere else. Or I could just keep calling it Turkey and move on.
When I have friends like that (well, my brother too), I make a point of typing it the way it's supposed to be written, out of courtesy and fun and proudness, like I'm clever enough to type any character I want with my keyboard, because I master it and I master other languages. The day I'll learn how to type Chinese characters fluently, I'll be so happy ! Not to boast, but it will mean I unlocked a whole new world!
if you want to learn to use endonyms for countries cool! but most people aren't interested in that level of effort for the whole world. there are ten thousand other things that occupy their minds, every one jockeying for a spot, and this being a passion of yours doesn't change that.
idk why you think there should be any different rules from this, except the general sense of anti-Americanism that seems to pervade this discussion.
Now, there is nothing wrong with calling it Türkiye in your own everyday speech, but it is not any kind of sign of good manners - it's simply a matter of personal taste. You'd still probably call their language "Turkish", while they call it "Türkçe", and the people living there "Turks" or "Turkish people", when they call themselves "Türkler", and even if you didn't you'd probably use your own grammar to conjugate these words as necessary etc.