Yes it does. You have to pay your taxes in U.S. dollars, so people have to use it. That's one of the purposes of taxes: so that the government-issued currency is used.
Of course, banks know this (it is the core of their business) and will therefore denominate loans in USD to avoid the risk of an unfavorable conversion during such a proceeding. Such an unfavorable conversion is exactly what the creditors in the MtGox bankruptcy are facing, because the proceeding involved converting their BTC accounts into JPY according to the exchange rate in 2014. Due to the change in BTC prices recently, that means that under the law the creditors will be fully repaid using what BTC assets MtGox still has, despite the fact that the creditors would receive a far greater payout if they received the amount of BTC that had been in their accounts at the time of the hack.
Beyond debts, there are also torts. Suppose I crash my car into your house and you sue me for the damage I caused. The courts will order me to pay in USD; so again, the courts are creating USD demand by creating an obligation to pay with USD.
So the demand for USD is only indirectly due to demand for goods and services; after all, there must be some reason why merchants are pricing their wares in USD. The direct driver of demand is the law itself, hence "fiat" currency.
That's exactly what I was saying: the goods and services are priced in USD because the legal structures require USD. I just used taxes as the most simple example.
But the whole point of the legal structures requiring USD is to get people to offer goods and services for sale in exchange for it. That way the government can purchase goods and services from the private sector to use as public goods, without having to first "get the money" from somewhere else.
If you want a challenge, try explaining why the Somali dollar still held its value despite being a fiat currency that lacked a government to back up its value for well over a decade.
Have you ever asked yourself why they believe that? It's because they can pay their taxes with it, but also, as it says right on the front, "this note is legal tender for all debts, public and private." As long as the US government is still a thing, this phrase means what it says. I haven't seen anything so far to convince me that BTC has more stability or longevity than the US or EU (or Britain...).
The dollar bill is worth 1 USD because the government asserts that.
That doesn't make 1 USD anything. It just means the dollar bill is worth 1 USD.
The government has no power to prop up the value of the USD if the world decides it's worthless.
I don't see "the world" deciding USD is worthless anytime soon: people in third world nations keep pristine US cash in their closets as a store of value; a lot of international trade uses dollars as a common currency; etc. If "the world" decides that dollars are worthless, that will be the least of my problems while I'm cowering in the bomb shelter I don't have.
Same for BTC.
It doesn't come from either of those things. It only has value because people believe it does, otherwise they would all try to get rid of it as quickly as possible.