Functional Finance says you only use taxes if you want to reduce Aggregate Demand to prevent inflation. The Long Run Government Budget Constraint says you use taxes to pay for past, present or future government spending. They sound very different. They aren't.
There's a general principle in economics: first you eat the free lunches; then you look at the hard trade-offs. Functional Finance says "first eat the free lunches". The Long Run Government Budget Constraint says "then look at the hard trade-offs".
You would print money and spend it. Or print money and use it to finance tax cuts. And you would keep on doing it, more and more, until you got to the point where you did start to worry about inflation. You first eat all the free lunches.
That's the underlying kernel of truth in Abba Lerner's Functional Finance (pdf) [see also his book The Economics of Control (pdf)]. And I don't know of any mainstream macroeconomist who would disagree. You use taxes only so you don't have to print money. When you get to the point that Aggregate Demand is high enough, so you start to worry about inflation, you use taxes to finance past, present, or future government expenditure precisely because you don't want to print more money and make inflation higher.
Suppose inflation isn't a problem right now, because Aggregate Demand is currently too low. Does that mean the government should print money and spend it? Not necessarily. Print money yes, but instead of spending it on goods, or on tax cuts, it might be better to use it to buy back some interest-paying government bonds. Because even though inflation isn't a problem right now, it may be a problem some time in the future. So you can buy the money back in future, by re-issuing the bonds (and save on interest in the meantime) without having to raise future taxes or cut future spending.
Here's an easier way to think about it. Print enough money to get Aggregate Demand and inflation where you want it to be. That's the free lunch the government can eat. Any additional government spending must be paid for, sooner or later, with taxes. The present value of taxes, plus the present value of newly-printed money (seigniorage), equals the present value of government spending, plus the existing debt.
Seigniorage revenue belongs in the government budget constraint. The government can pay for part of its spending by printing money. But don't get too excited. It's not that big, on average. If central bank currency is around 5% of annual nominal income, and if nominal income is growing at 5% per year (3% real plus 2% inflation) then 5% x 5% = 0.25% of GDP. (In the right ballpark for Canada -- it shows up as the profits the Bank of Canada hands over to the government -- but maybe double it for the US). Printing money is a nice little sideline, but we are still going to need taxes.
Let's ask a slightly different question. Why do governments pay interest on their debt? Actually, it sounds like a different question, but it's really the same question. Why finance government deficits with interest-paying debt, when you could use non-interest-paying currency?
The answer is the same: you pay interest on the debt to encourage people to hold it and stop people spending it. If you cut the interest rate on government debt, will people sell it back to the government for money, spend the money, and cause inflation? If not, then Aggregate Demand is too low, and the problem is deflation, not inflation. So there's a free lunch from cutting the interest rate on government debt. And the government should eat that free lunch. And then the Long Run Government Budget Constraint kicks in again.
Now suppose the real rate of interest on government debt is below the real growth rate of the economy. (Or the nominal rate of interest is below the growth rate of nominal GDP -- same thing). And suppose it will be like that forever, if you keep on doing what you were planning to do. The government can run a Ponzi scheme forever. It can borrow and spend, then borrow to pay the interest forever, and the debt grows more slowly than the economy, and the debt/GDP ratio declines over time. The Long Run Government Budget Constraint is undefined. The Present Value of taxes can be less than the Present Value of Government spending.
That's another free lunch that needs eating. The economy is dynamically inefficient. The economy wants a Ponzi scheme. And the government should satisfy that demand. Issue debt until the interest rate equals the growth rate. Then the Long Run Government Budget Constraint kicks in again.