Back in the day of integrated real-time keyboard-printer combos, a.k.a. typewriters, there was no way of typing a proper umlaut on Swiss IRTKPCs. You could type a double quote ("), then step back and type the vowel (or the other way round), but it looked pathetic.
On the other hand, Ae / ae, Oe / oe and Ue / ue actually had been the original way of writing an umlaut since the Middle Ages; around 13th century the "e" was moved on top of the main vowel, where it gradually turned into two vertical slashes and then two dots around the 16th century. Ae etc. kept being widely used in many scripts way into the 19th century, and so did the little "e" diacritic.
Since ä, ö and ü are used much more frequently than the capital variants, it made some sense in the entire 20th century to put them on the cramped Swiss IRTKPC keyboard but replace the Ä etc. with the French and partly Italian and Romansh accent diacritics.
Because Ae etc. were so widely used in typewritten text, most Swiss Germans do not even notice if someone writes Österreich or Oesterreich , and many type " Oel " out of a pure habit.
The eszett is a different matter. It originally is a typographic ligature of a "long s" ("s", also to be found in many old text in other languages including English) and a "z". It underwent several changes of its usage since the early Middle Ages and eventually ended up as a means to mark the different pronunciation of Busse (with a short "u", meaning busses in English) and Buße (with a long "u", meaning fines).
The same applies to Masse (mass in the physical sense) and Maße (measures, dimensions), Schoss (= Schössling , a shoot on a plant) and Schoß (the lap of a person), and, funny enough, Floß (a raft) and Dental Floss (which is English anyway.
Those are pretty much the only examples where the eszett really matters these days that come to my mind right now. But they are not important because the different pronunciations can be easily told from the context.
So, in the first half of the 20th century with the wide spread of typewriters, the Swiss said, "Hey, what the heck? Why should we keep using a special character just to avoid four pretty rare ambiguities of pronunciation? We have enough troubles with our other national languages, so let's remove that typographical appendix. After all, there are many more ambiguities in German orthography and pronunciation to deal with."
And they did. Very painlessly. The Germans, Austrians etc., however, stuck to it even through the latest Orthography Reform of 1996. I still wonder why. Not a single person in Switzerland has ever been hurt because the eszett got dropped, but because of four (FOUR!) ambiguous spellings you still have to use a character that exists only in one language and is a major p.i.t.a. to find on a non-German computer.
And those very same folks in their Ivory Tower force us (including the Swiss!) to write auswendig and aufwenden , but aufwändig ? Come on!