I disagree. Big Government means the government getting into every subject possible. The opposite of this is that you have a government that only enters into essential matters for the well-being of society.
Which has poorly defined boundaries because people disagree on essential.
In practice "Big Government" is used only for PR value. It sometimes seems the more a politician or policy wonk uses it, the more they are actually likely to increase government spending when they are in power.
Most people seem to use "Big Government" to mean they don't like something the government is supporting or regulating, and "Essential Government Function" to mean something they want the government to pay for or regulate.
A void of power never creates a favorable outcome for the common person. If there's room for commercial activity in the area, you can expect all sorts of unethical tactics to eventually surface, like we see in cryptocurrency. In a free-for-all capitalism people will lie and scam and cheat for profit.
So I don't really know what could the government get out of that would make things better.
We (the "owner class," parrots, what have you) would argue for the government regulating itself, not the companies. Facebook showing you fursuit ads as a result of your history isn't as big of a problem as the government showing up at your door now that you've shared the wrong idea online. Governments should not be allowed to indiscriminately collect data from (that is, to spy on) it's citizens. Giving the governmemt more power doesn't seem like the right solution here, being that that is exactly what we are trying to limit. Data should not be shared to other countries (like GDPR) and in the US itself it should not be collected by the government.
I've never heard anyone decry "big government" in response to governmental actions that they like, so I don't think it's usually a principled position.
Whenever people complain about Big Government I like to remind them that any attempt at regulating industries like Big Tech is decried by the owner class as "Big Government", which is often parroted by conservatives, libertarians, and other pro-business groups.
So is it Big Government when it protects the interests of Big Industry to the detriment of its country's people, or Big Government when it regulates it to protect the interests of the people? They can't both be the same thing, and I would argue what most people call "Big Government" is what they've been told to call it, namely the regulation of corporate interests. Nobody ever calls the consistent erosion of people's individual rights "Big Government".