I’m sorry but the logic isn’t too sound. That’s not how evolution works. Animals communicate with sounds, which isn’t proof either but at least as much an argument as the given one.

What convinced me is neuro linguistics: the brain region responsible for processing speech is next to the one responsible for hearing, also for sign language. This indicates that the former developed out of the latter.

Also: the idea of sign language being older than spoken language is an old one with quite a racist background: Plain Indians have a sign language as a lingua franca which is quite cool! Nothing is more neutral than changing mode and have a lingua franca that’s alien for all spoken languages. The European arguement back then was that “these savages” are on an intermediate stage were they have both the “primitive” sign language and the “advanced” spoken language as “we civilized people”. Again, this isn’t an argument that spoken language came first but I’m not convinced of the opposite.