I think your answer is good for speech but it is overly specific.

OP asked about language, this answer is only about speech.

There are good theoretical arguments that the first languages were in fact sign languages (the gesture-first hypthesis) https://link.springer.com/article/10.3758/s13423-016-1117-3 . This article is quite accessible and lists some of the main arguments for it (while ultimately coming to a conclusion that the author things they arrose at the same time).

One argument I remember but that that article didn't mention was the benefits of gesture during hunts and ambushes though, co-ordinating these with spoken language doesn't work all that well unless the hunter is already afoot. And we do see hunter gatherer societies with sign languages (or rather, it is the same language using sign) today still.

Other arguments that are in the article include primates already existing gestural systems, as well as the ability and capacity to teach them some symbolic gestures from sign languages (no, koko did not 'know sign language', but she did know some signs).

As we see this in our modern primate cousins, a greater affinity for signed languages as compared to complex vocalisations, I'd ask why would our ancestors be different? Given the advantages to signed languages at particular times, the developing hand and tool manipulation skills our ancesters were fostering, and the likely already existing gestural system, I'd agree that the gesture-first hypothesis is at least plausible.

Thanks. I'll check that out