I don't think there is a clear cause and effect (or even separation) of those two things. The early views/situations of those who made America are largely the same force which drove the desire for small government and vice versa. I.e. what you believe is ideal in reality is going to be based on the way you weigh things morally just as much as the way you weigh things is going to be based on what ideal is held.
There is, of course, more than one reason people hold the views they do, but I do believe things like rugged individualism & the Protestant work ethic definitely played a major role in why the majority of Americans hold the desires and fears about government they do - particularly in the aspect of these examples. Without those influences, I think what America has tried vs what it has feared would be a very different list today (which is not to say whether it'd've been more successful or not).
Did you know that you can be sentenced to life imprisonment in the UK for suggesting that the monarchy should be abolished? [1] That's quite progressive, since it used to be the death penalty. Now in reality you probably won't be, but that law is an echo, still ongoing, of the sort of systems everybody used to live under. That directly leads to ideals of a smaller government. So I don't think these sort of values we're speaking of are abstract or moralistic in nature.
It's also not just in the past. We continue to live in the era of mass domestic surveillance, Patriot Act, Snowden leaks, regular oppression for political views (see: pro-Gaza protests), two tier legal systems, and much more. All the while governments with vast resources plunge sizable chunks, verging on a majority of discretionary spending, into militarism flirting with WW3 and the viable end of humanity it would entail. And I'm not even going to get into the insanity that's going on, worldwide - including the West, with the treatment of political candidates and elections. Suffice to say that's also enough to make a lot of people quite concerned.
So again all of this makes one begin to think 'perhaps governments shouldn't have this much power or wealth' at a point that I imagine you would say is, itself, already relatively low. I think that should all be fairly reasonable, even if one might disagree at some point. But I'm not entirely sure what the comparable evidence for the alternative is.
[1] - https://www.theguardian.com/uk-news/2013/dec/13/calling-abol...
Well, again, it's not that these things take place of each other it's that they influence each other. Taking the first example, America only exists because individuals went to settle and conquer new lands and then pulled themselves up by their bootstraps to defeat the tyranny of the monarchy. There is no way in which this can happen without these cultural traits in the same way there is no way it could have happened and not driven these cultural traits further. The other examples follow the same line.
That's not arguing small government is supposed to be good or bad, it's just saying what the American mindset considers important when judging things - which is on traits which align with its mindset over its history, not on whether or not the application of policy seems evenly distributed to all.
I'm not going to debate the history beyond observing that most people were fleeing persecution or seeking a better life for themselves and their families. Scarcely few set out to conquer anything. But I think the most important point is that now a days I think if you put just about anybody in the current state of America, you'd see a trend towards desiring smaller government.
The exact phrase 'smaller government' is triggering because it's been so politicized into being a left vs right concept, but when left leaning individuals seek to decrease the strength of ICE, decrease the rights/capacity/scope of law enforcement, limiting executive powers especially as related to war, and other such things they are themselves also directly appealing for a smaller government, in at least some ways.
I think this hypothesis (that 'anybody' would trend towards seeking smaller government) is even testable with things like Asian Americans. Many/most have roots from more social type societies, and predictably trend towards voting liberal in America as a result of such, to say nothing of historical issues. But they are leaning more and more conservative over time, to the point that at current trends they will likely be a majority conservative bloc within a few decades.
It's simply that once one sees what 'big government' amounts to in the US and the current state of the US government, it quickly leads many people to start running in the opposite direction. And once you start running from such, the opposite (self responsibility/reliability/etc) inevitably grows in proportion simply because there is no other choice.