Foreign officials ignore the umlaut when looking at your passport or when transcribing what you scribbled onto a paper form. It's (rightly) viewed as a modifier of a letter they know. (In contrast, the sharp S is a letter they don't know. Many people think it's a B.) When filling in electronic forms, you're typically better off leaving off umlauts.
Source: A lifetime of experience with funky accents in my name, not living in the place where the name comes from.
That’s actually the worst thing you can do. That actually violates the relevant DIN standards for transcription.
You have to replace Ä with AE, Ö with OE, Ü with UE.
You won’t find a person named Müller under Muller in any database, especially not digital databases.
For electronic forms, always use äöüß, or, if the system happens to be an American one (the only form I’ve ever had to fill where I couldn’t use äöüß was a US DoD export declaration), use ae oe ue ss. Although be aware that some US software will have an issue with that, too, and just remove ß and assume äöü to be aou.
The machine-readable part of the German passport uses this DIN standard, so to avoid having to explain German orthography to some suspicious border guard, you'd better make sure you use it.
But when you get a US visa, the name in both the visual and machine redeable part is written with ä replaced by a, so you end up with three different spellings in your passport. And then you get a Russian visa, where they use a pretty strange transliteration into the Cyrillic alphabet (not the one I would use) and then transform that back to ASCII in the machine readable zone for variant 5. And so on.
I see your point. But wouldn't this problem be better addressed with an additional field on one's passport, which states the name in a standardized alphabet, e.g. ASCII?
Asian passports seem to include something like this.
Most (all?) countries that use non-Latin alphabets include a romanized name for these reasons.
The only problem is that often there isn't a good clearly unambiguous set of rules on how to do that. For the forms, it doesn't matter, so long as the country chooses one and sticks with it. But it can mess up the pronunciation of your name real bad.
The passports contain such a field in the machine readable zone. It has created some confusion in the past, as the transformations ß -> ss and ä -> ae are pretty surprising if you never heard of that.
Source: A lifetime of experience with funky accents in my name, not living in the place where the name comes from.