No that is not how money works, it is one way money can work but it isn't how money works. Many economies are debt free, because there is no reason to go into debt to have money.
Money is debt. Green (or whatever color) slips of paper aren't intrinsically valuable. They're a promise to redeem them for things that are, same as an IOU.
I take your point that not every economy is run by a government that owes 110% of GDP. But Japan and China are, and France and the UK are at 99%, so that doesn't seem to be the whole secret of America's success.
Money is debt. Even gold-backed money. Possessing money is the same as lending your time, labor, or assets to everyone else. To be repaid this debt you issued, you can in turn demand time, labor, or assets from someone else.
@ Jensson: As the others have said.. You might mean something else with your statement, but you cannot escape that the very definition of money is owed debt: A bank note is in essence an IOU: "This represents value you can redeem later".
You may have some idea you want to communicate about avoiding 'having a lot of debt' or owing more than you are worth, but money in human history has always been a mechanism of debt in its very nature. It represents being owed something else of value.
Isn't it better to have an economy based on debt than something people actually want or need? E.g. if we had a currency based on land, housing prices would soar even more than they already are at.
There are also downsides to it. One of the benefits of debt is to have more liquidity. If nobody owed anybody anything, but at the same time it was very difficult to trade with anybody, you would have a very inflexible situation that might not really be preferable.
Like if I'm sitting with goods you want, and you sit with goods I want, but we have no way to swap them, then we are both worse off.
The gold standard is not perfect but it's definitely better than what we have now.
My ideal pick would be using the price of a basket of common commodities (wood, iron, wheat); pay the bank for storing your goods and don't let state actors & friends steal your money with inflation.
I don't think BTC is the solution (it's unbacked) but wtfhappenedin1971.com has some good reading material.