Ignoring the political implications of the question for a second, if nothing else when people ask to be referred to by a different name, their chosen name is usually at least reproducible on the keyboards in common use. Most people in the US would have absolutely no idea how to type Ü, let alone İ. In addition, English speakers don't have [y], [c], or word-final [e] in their phoneme inventory.
Yeah, I fully expect people without easy access to u with umlaut to just write u (regardless of case), and those without easy access to capital dotted i, to just write a dotless capital I. And the pronunciation will be brutal, at least for the first few decades, but change is a process.
