You could actually pay for everything by just creating new money, though. It would inflate the money supply by the amount the government spends. I don't know if that would be enough to cause an inflationary spiral to where the currency would become worthless.
Government spending actually stimulates the economy by something called the multiplier effect. For every dollar the govt. pays an employee, for example, that dollar spent and re-spent locally turns into 1.6 dollars or something like that. This only applies for locally spent money though. If the workers send their money 'back home' or the govt buys tech from Europe then this doesn't apply.
Printing money exactly creates inflation, resulting in increasing devaluation of the currency. Following WW1 Germany printed money like crazy which resulted in hyperinflation. Eventually workers were literally bringing home their wages in a wheelbarrow.
Government spending actually stimulates the economy by something called the multiplier effect.
More so than if the government left that money in the economy in the first place? Taxes aren't black hole for money; the government just isn't very efficient at spending it. For instance, if the government pays out the money to a fat, lazy employee who does nothing of value, it enters the economy when that money is spent at Pizza Hut but an opportunity was missed for that money to go to an employee in the private sector who hopefully builds something.
It's been quite a few years since I took economics so I only recall a scattering of the details behind the concept of the multiplier effect of govt. spending. My point being that I'm not equipped to defend it or debate it, so you may very well have good points. Not to mention the actual value of so-called economic theories is open for debate. It's not like economists have had a great track record after all.
I do think, however, that it's probable and reasonable to assume that there's value in money circulating in the local economy. Certainly govt. spending is not 100% efficient as you point out, but all you need is it to be better 51% of the time for there to be gains over the longer term.
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