When we talk about the military as an institution it's important to distinguish between the subjective motives of people in the military and the objective function of a military within a particular society.
One of the underlying characteristics of the United States military is the presumption that it exists to advance "American values" around which there is broad social consensus: values like "freedom," "liberty," and "democracy." Though many who join the military do so owing precisely to a lack of "freedom" in economic terms, even these individuals can be counted amongst a larger group who accept this ideological premise, more often than not. Because military service is rewarded in cultural terms on the basis of "service to society," there is a disincentive to call into question the basis of one's reward: if the objective function of the military is likely to undermine subjective preferences about why people serve, it will not be examined.
Whether or not the US military advances "democracy" around the world is a matter that can be evaluated independently of the fact that it makes this claim. The criteria is straightforward: if the US military conforms to the preferences of the societies it impacts, it may be possible to make such an argument. It is immaterial here whether the US military conforms to the preferences of US citizens. Only those primarily affected by a particular authority are in a position to legitimate its rule.
Generally speaking, modern militaries are not constituted in such a way that makes them democratically accountable to the communities they affect. National militaries primarily exist to impose the authority of one nation on that of another. Because the US military is not exceptional in this regard, we begin with the assumption that the US military does not function to advance democracy, but to instead project internationally the authority of whatever groups presently hold US domestic power. If we incorporate into our discussion a class analysis of US domestic power, we will quickly discover that it diverges in important ways from any democratic distribution itself.
Accountable only to a small minority of US domestic interests, is it not more honest and respectful to veterans and service people to describe the US military as an obstacle to democratic aspirations, and to thank them for their efforts when they try to address this? In listening to them carefully, we may be surprised to find that they often do.