Your idea is reasonable, but I don't believe you have it the right way round, if there is a right way round.

I did not try to answer that question because it seems too poorly defined. Consider: innate sound signals and gesticulatory signals occur in organisms with no sign of flexible abstract communication. By the time you are dealing with organisms with more elaborate brains, such as monkeys, there are few gesticulations apart from innate, stereotyped threat, play,  or submission signals etc. However, oral signals, including in monkeys and chimps, are common and often culture-bound. Chimps may use more symbolic, conscious, intentional gestures, but for all I can tell, oral signals remain dominant.

However, all human cultures have their own gesture signals, even though they may be far less varied and rich than oral language, which is a pity. If you read Oliver Sacks' "Seeing Voices", you might be surprised to see how much we miss by neglecting the gesture channel. It certainly shocked us!