As someone who has a name containing a non-ascii letter living in the US, I greatly prefer simply removing the dots or apostrophe or whatever and just use the ascii version. Those transliterations like ü->ue or ø->oe cause much more problems than they solve. Yes, there are lots of letters for where this doesn't work at all, but for the ones that do, the principle of least surprise is better than phonetical correctness, imo. I pick my battles with Anglo- and US-centrism and this one isn’t worth fighting for, imo. “Whatever the letter looks closest to” has a much higher chance of a consistent interpretation. This isn’t even the fault of anglocentrism – it’s an issue between any languages with different alphabets. Heck, I don’t know Cyrillic, but at least with say Vietnamese i can go from ê->e without even turning on my brain.
EDIT: oh, and reconstructing a word or sentence with missing umlauts etc is very easy to do, at least in my language. You can infer by context if there are ambiguities in 99% of cases.