Most schools of thought suggest that Sign Language is a language, hence its descriptor. Signing shares many elements with spoken language and those who use it consider it a language. In the United States, many states recognize it as a second language and it is taught at some universities with credit as such. Most of society does NOT consider it a language on par with spoken and written languages because of the function of signing.
Signing may compare more to dialects of other spoken languages at one end of the spectrum and a paralanguage (more a pseudo-language) at the other than a language because it is not independent of a base language like English. Suggesting so is not off-based.
A paralanguage is a component of meta-communication that can modify meaning, give nuanced meaning, or convey emotion, by using techniques such as prosody, pitch, volume, intonation, etc. When people use American Sign Language, it stems from a mother language such as English and does not have its own generative uses and versatility like a spoken language. Qualifiers that signing is its own separate function outside of a base language abound, but as people speak and signers translate that speech into movements, facial gestures, and signaled spelling, it presents as a complex meta-communication tied to a mother language or base language.
It is not simple gestures, body movements, and hand motions with meaning because signing, like language, is generative in gestures and movement. But under the definition of language in this article, sign languages are more not language because none of them exist independently as do spoken languages.
When a person needs to communicate in the written word, sign languages use a base language to do so, English, French, and such. Using the English Alphabet is not a disqualifier since many other languages do so, like Xhosa. When the language is written, however, Xhosa generates its own words and grammar, not written English. With any sign language, there is no distinct written form. No matter how many dialects of sign language there are, in an English-speaking nation, signers communicate using English as the base language when written. That disqualifies it as a language in comparison to French, English, Xhosa, and others.
Attempts to make sign languages stand on their own are monumental tasks that have little practical value since there are not enough people with hearing impairment to support such a movement. It is neither convenient or practical to form a language separate from the base language when signers use a base language. It can occur, however. Signing can emerge as a language if the effort to establish it as such occurs.
Deaf, blind, and mute people communicate using braille and sign languages, but without a base language, could not function in society. It is unpopular to think of having a disability that places people at a disadvantage in society such as mutism or non-hearing. Communities exist for the non-hearing and the mute. These communities, however, function because most people in society are not disabled. Signing is for the benefit of helping people with disabilities communicate and live within society as equals though they have disadvantages. Signing works the same as spectacles or eye-glasses for the hard of seeing, hearing devices for the hard of hearing, prosthetics for missing body parts, or pacemakers for those with heart problems.
Argument could arise that language itself is a tool, a crutch to help unevolved humanity learn to communicate because of its lack of empathic ability generally towards understanding that would eliminate unintentional offenses. That is the subject for philosophers to ponder.