You've compressed massive time depth ranges into a single concurrent stream of events, which is not really what happened. Let's look at this from two angles, the first being human migration and the second being linguistic development.

Human migration:

The very earliest that people would have reached the Americas is around 35kya (kya = thousand years ago). This is not widely accepted, and the more likely age is between 15-20kya. It's not possible that the population of the Americas occurred any more recent than that. We can know this because the Monte Verde site at the southern end of South America is dated to 18.5-13.8kya, with the 13.8kya date being more generally accepted. It's not possible for people to have shown up earlier than that because they couldn't have gotten there by then. No one was coming by sea and landing in Chile.

Additionally, Beringia was submerged starting about 15kya so there was no overland route they could have taken anyway. So let's just say 15kya and go with that. That's the most generous as well for trying to line up with the linguistic records.

Linguistic histories:

Let's start with Asian Languages to mean Chinese because that's what you have in mind from what I can tell. Though really we need to go back even further, all the way back to Sino-Tibetan, the family that Chinese languages belong to, but that also includes Burmese, Naga, Jinghpaw and loads more. This past year there have been two papers (1, 2) put out that date the origin of Sino-Tibetan (hereafter "ST").

Now, keep in mind, we can not go back any further with these specific languages because ST is the origin, and we don't have any evidence of it being related to anything else (because the further you go back the more impossible showing these connections becomes). One of these two papers (Sagart et al 2019) puts the start of ST at 7.2kya. The other (Zhang et al 2019) puts the expansion of ST at 6kya at the oldest. These are both preliminary and I'm sure we'll see more and better analogies coming out in the coming years based on these models, but for now let's just take it at face value.

The calculation is pretty straightforward: If we accept the oldest of these two and say 7.2kya is reasonable, and we take the most recent expansion into Siberia as reasonable so 15kya, what we're looking at is that, literally as old as Sino-Tibetan is now (which is so old we can almost say nothing about it), the age of human migration into Siberia is over twice that age.

So we would expect absolutely no influence whatsoever of Chinese on languages of the Americas, because Chinese didn't exist until thousands and thousands of years later. The writing system is only attested back about 3.5kya, and that's also being pretty generous. Chinese characters didn't exist before that point, at least as far as we can know. That's also about the time of the first known use of writing (Chinese was a little later than that, but barely). So there should be no evidence whatsoever of written language from Asia showing up in the Americas having been brought by the migrations because writing didn't yet exist at that point (again, as far as can be known).

Other stuff:

On top of that, we have a few other issues to deal with regarding the migrations and linguistics, namely that we don't know what any language looked like at 15kya and what was being spoken by the people moving into the Americas. There's no reason to assume they were also only speaking one language at that time.

What we might see, however, is some contact between people who remained in Siberia and those who remained in what's now Alaska, except that there's a good chance multiple migrations happened, and that the populations in the NW northern hemisphere are not actually representative of earlier waves of migration across Beringia.

tl;dr: The timelines don't match up, no language existed that you would consider Asian, and certainly nothing Sino-Tibetan. That wouldn't happen until nearly 8000 years after people had already left Siberia and the land bridge disappeared.

References:

• Sagart, L., Jacques, G., Lai, Y., Ryder, R. J., Thouzeau, V., Greenhill, S. J., & List, J. M. (2019). Dated language phylogenies shed light on the ancestry of Sino-Tibetan. Proceedings of the National Academy of Sciences, 116(21), 10317-10322.

• Stoneking, M. (2016). An introduction to molecular anthropology. John Wiley & Sons.

• Zhang, M., Yan, S., Pan, W., & Jin, L. (2019). Phylogenetic evidence for Sino-Tibetan origin in northern China in the Late Neolithic. Nature, 569(7754), 112.