I've heard it commonly said that the Yellow River Valley civilization reaching back at least 5000 years was the inception of Chinese culture & history, birthing the mythic Xià dynasty followed by the much more concrete Shāng.
That being said, most sources that I've found make the claim that Chinese (in one form or another) is just a bit older than 3000 years old, as evidenced by the earliest known examples of written Chinese that we've found.
All of that being said, surely the Yellow River Valley civilization spoke some language, even if they didn't write it down. & if China came from that civilization, it seems reasonable to me to suppose that they would have spoken some archaic form of Chinese. I mean, why would a relatively contiguous culture suddenly change languages if they didn't have to?
If I'm not mistaken about what I've said here, might that mean that there is good reason to believe that spoken Chinese is older than we have direct evidence to support?
Yeah we don't know. There's no documentation that gives us enough to say. "Yellow River civilisation" is also a problematic term, since it basically covers multiple cultures over the course of thousands years. It really only has meaning as "any civilisation that lived in the Yellow River Valley", but not as a single uniform civilisation.
surely the Yellow River Valley civilization spoke some language, even if they didn't write it down
Yep, but we can only speculate, and not all that accurately.
if China came from that civilization,
Which is a big if
it seems reasonable to me to suppose that they would have spoken some archaic form of Chinese
Not at all necessary. Civilisations change language. It happens. And language doesn't stick to DNA lineages.
might that mean that there is good reason to believe that spoken Chinese is older than we have direct evidence to support?
Not really. Unless some actual evidence arises, then it's not the kind of claim we can make, nor one that as linguists we should be willing to make. Dating spoken languages is incredibly difficult and you essentially need to rely on other factors, such as archaeological evidence, genetic evidence and written records, none of which actually definitively settle such a question, because none of them are inherently in lock step with the spoken language.
The estimates for the dating of Old Chinese are what they are because that's the best we can get with the existing evidence.