Is "but by the alacrity with which it got out of its way" by Thoreau a correct sentence?

Not correct, that is a nonsense sentence.

Governments show thus how successfully men can be imposed UPON, OR even impose UPON themselves, for their own advantage. It is excellent; we must all allow THIS. Yet this government never of itself furthered any enterprise, but got out of its way with alacrity.

The meaning of the next three sentences are not clear. Either the Government or an enterprise could be being referred to here.

It does not keep the country free. It does not settle the West. It does not educate.

Who doesn't ? Is he talking about the Government or about an enterprise?

Try this as a replacement: "The Government does not keep the country free, neither does it settle the West nor does it educate the people."

You can replace "people" with "population" if you like

Alacrity Meaning

Yes. The third sentence avoids ending the sentence with a preposition, it just says that the Government never helped any enterprise, just got out of the way quickly i.e. didn't impede it.

I am not sure what to say