Just to echo what marplots said, I think I should also offer my support. I'm certainly not against early infant learning and I am not trying to rain on your parade.
I'd just like to offer a few more pages of that book by Tomasello. This leads on from the previous pages I scanned. It's more than ten years old, so some of the research could be out of date by now, and it does follow a particular theory of language acquisition that not all linguists agree with (in particular, Tomasello's book is meant as an argument against Chomsky and Pinker).
However, in this chapter on early holophrases and early linguistic ability, he points out that we should not necessarily think of language as being built from phonemes, to words, to sentences, to paragraphs. Instead, children pick up on salient words and phrases. Some of the infant's learning will be combinatorial, but other learning will involve segmentation. For example, children will learn whole frozen phrases such as "I-wanna-do-it" or "You're-welcome" without realizing that they are made up of words. This process is segmentation.
For example, in English there are a lot of phrasal verbs. Children often first learn prepositions in a phrasal verb ("off" instead of "take off"). In Chinese, on the other hand, which has fewer (no?) phrasal verbs, the word "remove" will be more quickly learnt. Chinese has less (no?) morphology also, so infants will not have to go through stages of learning the morphology for past tense or plurals as English-speaking children do.
But also, he points out that children need "intention-reading" skills such as:
joint attentional frames
an ability to understand communicative intentions
role reversal imitation
And he says, "Language emerges in children in the months following the first birthday - and not before - because this is when these fundamental skills of intention-reading are solidly in place." [p.41]
So it may not be possible to teach actual sign language to babies prior to their ability to focus on people, understand that other people have some form of agency, desire to imitate etc...
I would think that if there are counter-examples to this then people in the deaf community would have discovered an early ability for actual linguistic ability.