C'mon that page is highly technical, really just listing the letters or glyphs used for forming printed text. In reality, if you walk into any first grade classroom you see pictures of the letters A-Z with pictures (Apfel, Bär, uwv) and then after the end maybe around the corner, what, Öl? I can't even remember. When the kids recite the letters they don't recite äöüß. TBH I really only remember this because when kiddo was that age the Neue Rechtscribung transition was mid-process and the parents were angrily divided so I was kinda paying attention.
Also, though it's hardly authoritative, my kids' school taught English through immersion from grade 1 too, and both German and English teachers said "same alphabet".
As bmicraft pointed out, even in that wikipedia chart those inflected letters are spaced apart from the others. Yes, they are letterforms, but not part of the "alphabet" -- they don't even have a sorting like the Swedish Ä or W do.
And you can switch in running text from using the marker for umlaut (dots or bar, not semantically dieresis) or a normal "e" without anyone blinking. There's no problem reading a Swiss book even though ß refuses to cross the border. Though I personally prefer to read Äpfel and Bär rather than Aepfel and Baer, really, they are the same.