I was certain I remembered hearing a theory that the Chenjesu are constructed, and upon checking the wiki I found it fairly quickly.
Their rare physiology, advanced sentience and peculiar evolutionary path leads some to speculate that they are the product of an ancient, peaceful culture. An alternate explanation is that they are mostly software, interacting with their environment through some kind of field manipulation mechanism.
And the source for that is apparently an IRC chat with Fred Ford and/or Paul Reich III:
<_Stilgar> <zenir> Q: how did the chmmr/taalo evolve.. especially to sentients?
<Fwiffo> Adaptation and evolution require the ability to change in response to the environment. The Taalo and Chenjesu must either be the product of a previous culture, or they must be mostly software with some kind of field [manipulation] ability.
So while it's true there's no in-game indication of it, there is evidence from the creator that they may be constructed.
As for the Precursors thing... the Precursors are the only ancient advance race that we know of who could have created the Mmrnmhrm. It's quite possible that there's another advanced race out there, but since we don't know about it we're more likely to suggest it was the Precursors.
EDIT: And yes, I was off by an order of magnitude with the Precursors: they lived between about 300,000 and 250,000 years before the events of SC2.