On my firefox (68.9.0esr), a Ctrl+F for "The German character a" finds no results.
Also, as a Finnish user, I would not be impressed by a search for "talli" (stables) matching "tälli" (blow), or "länteen" (westwards) matching "lanteen" ('of the hip').
In contrast, as a Swedish user, I'm spectacularly impressed when, say, a flight or train booking site lets me search for Goteborg rather than Göteborg. Because otherwise, you know, I probably can't get the fuck home from wherever I happen to be with whatever keyboard the particular machine I happen to be using has.
The number of times, in actual real life, that the inconvenience of the ambiguity outweighs the convenience of the overlap are not many.
I would assume that sites like this know these words as synonyms and don't simply do a lax string matching, because in this specific case "Gothenburg" would probably also work - it did when I was planning a route through Sweden with a German app that defaults to local spellings in foreign countries.
Flight and train booking sites need to support search for a very small set of place names: if they care about selling tickets they can list a synonym list for each destination, sidestepping language and spelling norms. Fuzzy completion in search forms as one types, like e.g. in Google Maps, is also effective.
"Should match" above was too strong. "Should match for users with locale set to English" is right though. In English, theoretically you should write "café" and "résumé" but practically speaking, it doesn't matter and no one cares if you omit the accents.
In the US, there's starting to be more push from people with ñ in their name to get support for the correct letter, but most government systems have zero support for anything but ASCII all capitals (sorry McWhoever).
I think it depends on the context. I need the search-and-replace dialog in my text editor to be strict about it, whereas I would find it weird if my web browser would not treat a, à, â, and ä the same way. After all, in the languages I speak they are all the same character, just with different accents. This is of course also language-dependent.
Firefox only gained that ability very recently in FF73 (https://bugzilla.mozilla.org/show_bug.cgi?id=202251), hence it's not yet in the current ESR. It does seem to have a toggle for turning it on/off, though.
Normalisation and - except for the soft-hyphen - ignoring of ignorable whitespace characters (ZWNJ, ZWJ, WJ etc.) on the other hand are still missing in Firefox (https://bugzilla.mozilla.org/show_bug.cgi?id=640856).
> On my firefox (68.9.0esr), a Ctrl+F for "The German character a" finds no results.
On mine (77.0.1) it finds the OP's "The German character ä" as well as yours. Conversely, searching for "The German character ä" finds the OP's original and your "a" version. There is a "match diacritics" button for enabling/disabling this behavior.
Also, as a Finnish user, I would not be impressed by a search for "talli" (stables) matching "tälli" (blow), or "länteen" (westwards) matching "lanteen" ('of the hip').