The history of sign languages is more akin to that of creoles than that of a constructed language which later gained native speakers. Consider the following two examples:
The grammar of Esperanto and a large chunk of its vocabulary was created, codified and written down by L. L. Zamenhof. People who were interested learned his language according to his rules and later passed it on to their children, creating the first native speakers of it.
The development of Tok Pisin is described as follows: Contact between English speakers and locals of Papua New Guinea gave rise to a pidgin, an incomplete and bare-bones not-quite-language that was just good enough to communicate - everyone would have spoken it a bit differently at that point in time. It became used more commonly and was passed down as a native language to children, creating a full-fledged language. At no point however did its speakers sit down and decide on what grammar Tok Pisin ought to have.
Now compare these with the case of the Nicaraguan Sign Langauge, the currently only natural langauge which we have actively seen develop. Deaf children of Nicaragua previously communicated with the people closest to them with improvised home signs and gestures (in other words they had at best something akin to a pidgin). When they were brought together in a new school for the deaf, they had the need to communicate with each other. The school encouraged use of mouthing or fingerspelling Spanish, but the students, who were not native speakers of Spanish failed at learning these skills. Instead, they started communicating by using their own home signs, forming a pidgin that developed over the following years and was taught to younger students who later arrived. Over time, the language developed complexities similar to what you would find in other signed languages - evidence that it had developed beyond the initial pidgin stage.
It is clear that the example of Nicaraguan Sign Language is much more similar to that of Tok Pisin than that of Esperanto.