SignWriting is closer in purpose to the International Phonetic Alphabet for spoken languages. It attempts to allow detailed recording of the actual signing as it is signed for any sign language.

It has a lot of the disdvantages of IPA as a practical writing system as well.

Sign languages are not the same as spoken languages used in the same countries, as is very apparent if you look at transliterations of ASL using latin glyphs, there are some standardized ways to do this but they drop a lot of information and don't have the same sentence/word structure.

There is also a long history of attempts to create notation that can record this type of language, the first for ASL being Stokoe notation, which represents hand shapes for example, but can't represent for example facial elements, and is specific to ASL, can't represent things in other sign languages.