Some of these concepts exist when limited to ASCII.
For example, in olden times, or when restricted to ASCII, the Nordic letter "å" is written "aa", but it is still sorted at the end of the alphabet — "Aarhus" will be close to the end of a list of towns.
In Welsh there are several digraphs, single letters written with two symbols. The town "Llanelli" has 6 letters in Welsh. (There are ligatures, but I don't think they're often used: Ỻaneỻi.)
Indeed, collation, case-insensitive string matching, and probably a bunch of other things must be used with an appropriate locale. That was the case before Unicode and is still the case with Unicode. The only difference is that the tables for how to do it are slightly larger now, but the operation itself isn't (much) more complex.
For example, in olden times, or when restricted to ASCII, the Nordic letter "å" is written "aa", but it is still sorted at the end of the alphabet — "Aarhus" will be close to the end of a list of towns.
In Welsh there are several digraphs, single letters written with two symbols. The town "Llanelli" has 6 letters in Welsh. (There are ligatures, but I don't think they're often used: Ỻaneỻi.)