Does this example of a bad government regulation come from the government? Or the people voting and demanding it from the government?
No, it comes from business. DMCA is an example of the badness of private business if allowed to do what it wants.
That was a purchased law. The failure on the governments part was in allowing itself to be for sale.
So you are right, even though I bet this is just about the opposite of the conclusion you thought citing the DMCA resolves out to, we should definitely stop allowing the government to be influenced by the private sector.
Look, it's government power behind it. I don't fundamentally care how it's gotten rid of as long as it is. Weaken businesses, weaken the government, weaken the links between the two, whichever way is fastest/most effective- in principle. I just don't particularly trust your prescription, as you don't mine. This situation is, of course, exactly as the government-business complex wills it.
There are actually quite a lot of government regulations without which a search monopoly would be less likely, from the patents Google holds to the CFAA creating uncertainty for small competitors to do the scraping that would be necessary to get as comprehensive an index without being large enough to have sites request indexing to the GDPR increasing compliance costs to the RIAA/MPAA and news sites flinging lawsuits left and right based on overbroad copyright laws to the FCC rules that impair last mile competition that could result in higher upload speeds and thereby greater P2P content hosting which is more easily indexed by search competitors.