Simply false. If interest payments can be made sustainably, there is no default.

At some point though you have to balance the budget or they won't be sustainable.

Nope, don't even have to go that far. So long as interest payments as a fraction of tax revenue doesn't increase, debt is sustainable. You can raise tax revenue either directly or by growing the whole economy, so the taxes are the same percentage, but coming out of a larger number.

It's no different than personal finance. If I get a raise and my income goes up by 10% and get a fancier apartment for 5% more, I'm still better off overall because my salary grew at a faster rate than my rent.

True, as I understand it though they still have a fair bit to go to bring it to a sustainable level in that sense.