How does any of this hold up logically?
Most of the time, it doesn't hold up, nor is it very consistent, as you and many others have noticed. A lot of people view the government in the same way some people are said to relate to the police: They hate them until they need them, and then they love them.
I mean, you can't BOTH want a small government AND have the largest military on the planet, which makes up a considerable amount of the US budget, can you?
It's not necessarily the same people or factions who want small government and the largest military on the planet. "Small government" may not be a very descriptive term anyway.
What people want in a government is not so much a matter of "size," but efficiency. Some believe that there's too much waste and inefficiency in government. The bureaucracy is too bloated and oversized, not to mention widespread corruption and too many government officials feathering their own nests. The late Senator Proxmire used to give out his "Golden Fleece Award," which would cite the most outrageous examples of government waste and inefficiency. Plus, government employees get a wide array of benefits, perks, and pension plans which their private sector counterparts can only dream of. (If every employer offered the same health benefits as the government gives its employees, Obamacare would have been totally unnecessary.)
Apart from efficiency, the other major factor in motivating "small government" is a matter of rights. A smaller government is less likely to violate an individual's rights. It doesn't have to be literally "smaller," but more restrained and less likely to hassle people. Most people prefer to be left alone and don't want to be bothered by an intrusive government meddling in their private lives.
As for having the largest military on the planet, it does seem that it would contradict the notion of small government. In addition, those who support a large military also ostensibly support a large national security state, large police departments, and an oversized corrections system. Their idea of "small government" involves cutting in every other area, such as in transportation, education, social services, healthcare, etc. As a result, our roads are strewn with potholes, our bridges are collapsing, our schools are graduating functional illiterates, our social services are abysmal, and our healthcare system has the highest price with the lowest quality.
But overall, the idea of "small government" is a pipe dream and a false promise. Fact is, the mostly conservative and capitalist business community has grown dependent upon government pork. Even in the military, there are some who believe that the military can still accomplish its mission without being so large. Some advocate a modern, yet "lean and mean" fighting force, as opposed to the behemoth we're now feeding. The Pentagon is often thwarted whenever they propose cuts by closing bases which are deemed unnecessary. But that's when Congress starts balking, as they don't want to lose the economic benefits of having a large military base in their districts. The biggest and highest-paying employers in my area are a state-owned university (which thrives on government contracts and grants), a federally-owned air force base, and a large military contractor.