[hnfong]

That's not quite what happened (or it is a overly simplified version of the historical accounts...)

First, it wasn't clear when the Tao Te Ching was written. Anyway, the political philosophy that survived and emerged "victorious" during the turbulent periods was the legalist school of thought, which was like Lao Tzu and Confucius married and had a child together. The Taoist ideas of Wuwei (non-action) morphed into the idea of automation via laws and regulations. Their logic was that rulers are not supposed to do the work themselves because it would not "scale". In the end, Qin, who went all in on legalism, having made the whole kingdom a terrifying economic and military machine governed by strict regulation (incidentally, informed by "behavioral economics"), conquered all the other states.

The Taoist ideas we know from Tao Te Ching only gained prominence after the legalist school was dealt a deadly blow due to the quick dissolution of the Qin empire. The new elite (that of early Han) thought that the harshness of strict laws and cutthroat competition were the cause of the empire's collapse. It was only then that the "take it easy", "small government" kind of Taoist ideas took hold.

[MadSudaca]

The first TTC manuscripts date to around 300 b.c. and there’s evidence that some of the poems have a rhythmic structure, which could mean some verses were transmitted orally prior to that. So there’s a good chance that it was known prior to 300 b.c. between that time and when legalism crumbled, there were other renowned Taoist sages, such as zhuangzi. So while legalism was still in full force, Taoist ideas were still evolving, granted with less vigor.

I would suspect the “take it easy” part took root due to the strictness of legalism. Indeed, there’s verses in the TTC that I see as a criticism to legalism, for example, 74.