No, plenty of things have value even though you can't use them to pay taxes. The kiwi fruit has value. Why? Because people want them, yet kiwis are not strictly speaking NEEDED.
Again, what gives value to things is that people want them, no matter what the reason.
But there is no difference between kiwi fruits or fiat currencies: people wanting them is what gives them value.
In the case of bitcoins, people want them not to pay taxes, but because cryptocurrencies offer benefits that only them can provide, so people see value in that.
Fiat currency is just a number. Who would want a number? It's taxes that create demand. The problem with the supposed usefulness of bitcoins is that the interesting part is the technology, but the numbers are irrelevant. You can make a transaction for 1 bitcoin just the same that you can make it for 0.0001 bitcoin. Now matter how you dice it, ultimately the number has no real value.
Taxes are just one small part of the demand. The economy makes up the rest of the demand, and it accounts for a much larger share. Even companies that are not US-based and do not pay US taxes can and do demand US dollars, eg. a Chinese company exporting goods to the US and receiving payments in dollars.
The US government income (federal, state and local taxes) was $1 trillion in 1988 [1]. I picked this year because it is hard to find up-to-date data for the next number: the amount of dollar transactions was estimated at $1.7 trillion per day in 1988 [2]; this is $620 trillion per year. So taxes represented only 0.16% of all dollar transactions. At most you can argue that 0.16% of the value of the dollar comes from its use to pay taxes, but the bulk of its value comes from the economy using 99.84% of it.
Still not convinced? Here is a thought experiment. What if for every dollar owed in taxes, the government asked to be paid instead 1 grain of rice? Would it suddenly make the dollar worthless? No! People would continue using it because many other people already use it to transact trillion of dollars every year. They know it is a liquid asset that can be exchanged for anything, and this is what gives it value.
Comparing the sum of economic transactions to government income is not relevant. Just as with anything in economy you have to think in the margins (marginal demand, marginal tax rate, etc.) I'm not saying that taxes are the only demand. USD has value! It doesn't matter if you are Chinese! I have another thought experiment: imagine that the US (or any other country, you can think of a small country as it will be more evident) decides to issue a second currency, but you can't pay federal taxes with it and there is no fixed exchange rate to the USD (notice that it's not a bond, it has no maturity).
In any case, there is no need for thought experiments with bitcoin. Reality will show you and me if bitcoin has any sustenance. I think it's an amazing technology, but bitcoins themselves have no value (in the long term).
"Comparing the sum of economic transactions to government income is not relevant. Just as with anything in economy you have to think in the margins (marginal demand, marginal tax rate, etc.)"
My comparison is very relevant. It means if taxes did not have to be paid in dollars, then with all else being equal there would be 0.16% fewer dollar transactions. Not a big change.
You talk about marginal tax rate, but I think you meant tax rate. (Marginal tax rate has nothing to do with our discussion). If tax rates where doubled, then taxes would account for not 0.16% but 0.32% of dollar transactions in my 1988 example. Again, not a big change. And yes in practice marginal demand of the dollar would skew this percentage somewhat, but it would still be insignificant.
You did not answer if you were convinced by my thought experiment, are you?
About your thought experiment: what you describe has actually happened many times in history, and many of these currencies had a floating exchange rate and could not be used to pay taxes, see https://en.wikipedia.org/wiki/Private_currency Turns out they had value, like Bitcoin.
A difference in the case of any currency, fiat or otherwise, is that a huge part of the reason people want them is an expectation that others will want them tomorrow.
