Why does the value of a currency derive largely from the existence of income taxes?

Because they induce a demand that might otherwise be satisfied by a different currency.

The low inflation in the US before income taxes were introduced falsifies the argument, no? No other currency was widespread in the US prior to income taxes. I still don't actually understand the argument, to be frank.

The US used to exchange gold for dollars which gave them inherent value.

That is no longer the policy and other currently existing currencies do posses the potential to compete with USD. Why would I choose to settle my debts in a rapidly-inflating currency that is backed by nothing?

Well, I agree that the combination of printing money with no taxation isn't going to work due to inflation.

I just didn't understand the statement that a currency's value "derives largely from the ability to pay [income] taxes with them". That statement by itself is historically false, as the gold-backed currency shows. It still makes no sense to me even without gold-backing. As long as the money supply is kept relatively stable, all you need is for a few government services to operate exclusively in USD (e.g. welfare, where funds are raised via land tax or carbon tax for example), and for a reasonable amount of trust in the government, and to perhaps kickstart the group delusion of trust with initial gold-backing, and you're all set. I can make sense of the statement strictly in the context of money printing, where you need the offsetting deflationary impact from taxation (not necessarily income taxation), but that meaning wasn't so clear from the original statement that I was responding to.

It's true that it's not strictly necessary that the "backing taxation" for this scheme be specifically on income, but due to historical circumstance this the largest pillar currently supporting the US system. Theoretically income should be related to productivity so it is a reasonable backing to balance against inflation. The onus is on those who'd like to replace the system to justify a better alternative.

Government debt; so long as that exists then the value of the currency will be derived from the ability to fund that debt through raising revenue - i.e. taxation. Without governement debt, different story.