To articulate the core issue one more time: I don't understand how U.S. armed forces protect democracy, or how they have any impact on the civil liberties/constitutional freedoms of Americans or citizens of other countries.
One would have to reach a bit to argue that the US military directly defends American civil liberties from outside threats, but for other nations freedom I think its more tangible.
The United States and its massive collection of nuclear weapons make up around three quarters of NATO's military strength, making it the main deterrent in the alliance. Given what Russia's doing in Ukraine right now, and the kind of rhetoric they're throwing around about the Baltic states (Estonia, Lithuania, Latvia) and Poland recently, it can be argued that similar things might be happening in those countries right now if they weren't guaranteed by NATO and its nuclear umbrella. This of course continues on from the Cold War where the United States guaranteed Western Europe's freedom with nuclear weapons and a significant conventional military investment.
For a more long term example, there's South Korea. It only exists today because of the UN intervention (the vast majority of which was US troops) and the US presence there, and the implied nuclear umbrella, continue to guarantee its independence. It's worth noting that South Korea was still a dictatorship for its entire history until the 1980s, however they were free from North Korean control which I'd argue is still a certain kind of freedom. But even putting that aside, the US does still guarantee South Korea now that it's a respectable liberal democracy.
Moving on but staying in Asia, there's a more indirect example with Taiwan. The US has no public defense treaty with Taiwan but has been deliberately ambiguous about what it would do in the event of a Chinese invasion. I think it can be reasonably argued that if the US became an isolationist power tomorrow, it'd likely be under Chinese control within a month.
And now, to get into an area that might be a little controversial: though I think the Iraq war was based on complete fabrications and a warped neoconservative ideology and wasn't worth it, I think it still merits pointing out that the US chose to "give them" a democratic government instead of installing another dictatorship. As we now know well, it's a complete failure because of nepotism, corruption, and sectarian struggle, but I think you can only blame that failure on the US insofar as the US was naive to think Iraq could support a functional liberal democratic kind of society after decades and decades of dictatorship and a complete lack of civic nationalism and trust in public institutions required for that.
The invasion of Afghanistan wasn't morally abhorrent like the invasion of Iraq, but otherwise the story is the same, except more extreme. There can be no comparison between Taliban rule and today's Afghanistan (at least in Kabul), even corrupt/dysfunctional as it is.
Another controversial example I'd argue for is South Vietnam. The Vietnam War was executed ineptly and South Vietnam was a terribly corrupt country, but at the same time it was still less shitty than wartime Leninist rule in North Korea (Vietnam isn't that bad these days, all things considered). Even if South Vietnam was a shitty regime that people were unhappy with, all you have to do is read about the Vietnamese boat people to know that they still had some freedoms they cherished that they were afraid of losing when the North came.
And then if we want to go back to the most blatant example, there's WW2 where the United States and its allies reestablished democracy in every nation it liberated, including Germany and Japan, the latter of which US military lawyers were kind enough to write a constitution for that I'd argue is better than America's own.
The US does still support many dictatorships, and his installed a lot of them over time, but these don't really involve the military. The overthrows are done by the CIA and the support for dictatorship is done through intelligence support, sale of military equipment, etc.
TL;DR - the US protects the freedom of a lot of countries in Europe and Asia by keeping China, North Korea, and Russia in check, even if the intelligence services and executive branch traditionally haven't in other places.