First correction of various misconceptions:
Since Nixon went off the gold standard the U.S. dollar has been a fiat currency backed by federal debt.
U.S. dollar is not backed by federal debt, and in general fiat currencies by definition of the word are not backed by anything (see Wallace 2017).
Also, generally if money is backed by something it has to be convertible/redeemable at some fixed rate for that thing. U.S. government does not give you promise to exchange your \$100 note for its bonds upon request. Moreover, if U.S. government would feel like running surplus budget they do not need to sell you any debt instruments either.
I assume that what "backs" the value of the fiat dollar is faith in the stability of U.S. bonds and the government that issues and repays those debts.
Again not generally correct, value of fiat money depends on demand and supply for that money. Depending on what sort of situation regarding government debt we talk about it might affect value of currency or not (see overview of literature on various aspects of sovereign debt in Reinhart and Rogoff (2009), some passages of the source on debt thresholds are outdated and no longer accepted but the overview of literature on sovereign debt crises and defaults is still valid and comprehensive and discusses cases when debt crises led to loss of value of currency and cases when it did not).
Regarding the main question
Is Fiat Money Ultimately Tax Based?
No and yes.
Fiat money by definition is not based on anything else than government decree. However, value of that government issued money will depend on demand for that money and supply of that money.
One important way how governments can artificially create demand for their money is by requiring taxes to be paid in money they issue. Taxes do not necessarily need to be paid in money, in principle they could be settled with in kind payments. However, if state requires you to pay taxes in dollars you always have to make sure you will have some dollars at hand when you need to pay them. Thus you will demand dollars even if you personally may not like them, and everyone else does the same. This creates non-trivial portion of demand for money and that is what gives money its value.
In addition, if government runs surplus and does not spend the money it collects through taxation it also reduces velocity of money in circulation which is tantamount to economy having lower quantity of money, so this would increase money’s value as well.
Sustainability of government finances affects value of money only to the degree it affects supply and demand for money. Moreover, as you can read in Reinhart and Rogoff (2009) empirically more often than not sovereign debt crises are resolved through debt monetization (which increases supply of money), and they are often accompanied by capital flight from a country (which both decreases demand for country's money and to increase in supply of money). This is why countries with debt problems typically end up with money that has low value, not because fiat money is in any meaningful sense backed by government debt.