A better comparison for the content of the SignWriting page on the English language WikiPedia page would be the pages for Arabic, Hebrew, Chinese, Japanese, Linear B, Klingon, Na'vi and other orthographic systems. We need to be clear that SignWriting is nothing more than a writing scheme for signed languages. In the same way that Chinese calligraphy is a writing scheme for spoken Chinese and similar for other spoken languages with non-Latinate orthographies.
Personally I consider SignWriting to be closer in purpose to IPA (International Phonetic Alphabet) than to written English. Lexicographers might disagree with me and suggest that Stokoe notation is the obvious parallel to IPA. Though here as a native English speaker and second language British Sign Language speaker I would contend that the reliance on ASL fingerspelling shapes names in Stokoe, and the use of Latinate symbols in IPA for that matter, are a hinderance to learning the notation. There is, of course, a similar problem with SignWriting as many of the training materials are written using ASL as an exemplar. The iconic nature of SignWriting allows one to get around the problem, which a non-Latinate reader would not be able to do with IPA. I could just as easily say that HamNoSys is the IPA of signed languages but the point is that the written critical form of spoken languages often bears no relation to the way strings are actually pronounced. (One only has to consider the lyrics of the Gershwins' song "Let's call the whole thing off" to see what a mess standard English orthography makes of pronounciation.)
However, in one sense I agree with you; the use of an ASL story gives the wrong impression of SignWriting ... that it is solely for ASL. As Val has pointed out (thanks Val for correcting my poor history of the genesis of SignWriting) this orthography is applicable to all signed languages and manual communication systems.
A number of people responded to my suggestion about signwriting (SW). Only two of them understood where I was coming from and why I made the suggestion that a simpler, more straightforward piece is called for in the Wikipedia article. Valerie Sutton mentioned the origins of SW and how it arose from someone without a background in sign language. I think that all of the respondents should read and think about what she wrote because at the time she was also not involved in SW as it has developed. The other person who made very pertinent remarks is Stuart Thiessen, who went through the same experience that I have, viz., very little knowledge at a very early stage of learning ASL. He, too, needed responses to questions that arose from very little experience with ASL