It may be nitpicking the point, but a lot of this is a non-issue in most cases and you're setting up a false dichotomy with straw-men.
The reason people don't think about a "wealth gap" with their government is multifaceted:
1. The government provides goods and services that they use and since most people gain benefit from it, they don't question the outcome or source.
2. A government as an entity usually doesn't run a surplus, so while they may have a $1T budget, they're running at a deficit, if they were a human, we'd question their spending habit, not be angry or envious. Wealth, by most people's standards is a positive P&L.
3. Since a government runs at a net loss, and provides a net gain to society there isn't a loss of emotional capital towards the government in most people's minds.
Banks, while they make a tidy profit (especially the large banks) don't have what you would consider large amounts of fluid capital either. They take and store your money for later, at some rate; they will also give out loans based on criterion that allow them to make a small sum off of the interest due to the assumption that most people will make regular payments (that's why you have to apply and just don't get the money). The large majority of banking in the USA is actually done through local banking corporation than big box groups, so getting angry at all banks because the folks at Lehman Brothers wear hats on their asses is a gross over generalization.
A standing army, much like the government that runs it, is removed from the capital equation since they are an entity that resides out of the normal checks and balances of a capitalist style society; this is why you see both military and governmental entities under any system that isn't fully Anarchistic. Police don't harass people due to the fact that they are funded, those that harass people do so because they're assholes, we care about people being assholes.
Your definition of capital is even too broad; capital generally implies all physical goods, whether that is money or product is rather immaterial, these things are there no matter the system, even Anarchistic systems have capital. Capitalism on the other hand is an economic system codified generally by the supply & demand curve that most people see in Economics 101; this can be contrasted with Communalism, Agrarianism, Communism, Bazaar/Barter, and various other economic structures. These are inherently separate from governmental systems that the lay-man usually associates with them, such as: Socialism, Democracy, Republics, Anarchism, Oligarchies, and the like.
It comes down to this; people care about others being jerks, it doesn't matter if you have money or not. If you aren't sensitive to the motives of those around you, whether you're doing it out of spite or ignorance, it will be taken like you don't care and spite/anger is a result. The article is pretty darn good at pointing that out, and I would tend to agree. I would also tend towards most people seeing someone with a wealth level that they only see from Powerball and the Lottery, complaining about getting taxed to be the most abhorrent form of pedant whining that could be imagined; or even worse, completely writing off the fact that the system is set up to exponentially reward people with enough capital to take advantage of it. Paul's expose is along those lines, he's defending a system that encourages the top 1% of 1% of our society to expand exponentially while the rest ride the wave and accept the scrap.
Are CEO's really doing 1000x the work as an engineer?
Are Angel investors really doing 100x the work of a founder?
Should someone playing the stock market really be taxed at a nominal rate nearly half that of someone earning a wage as an engineer, scientist, or even a burger flipper?