GB & Germany are getting closer to a balanced budget (Germany will probably get there next year). It's not impossible to pay off all debt, especially if inflation returns to healthy 3%.
The political term "balance the budget" seems to be a corruption of the accounting term, and it means just for the state to break even. It seems likely to me that the coiner of this term had some misconception about double-entry bookkeeping - such as that you break even iff your books are balanced
There is no need to pay off your student loans completely if the interest rate on them is low and you can do better things with your income. Making your payments is responsible. Paying them off (partially or entirely) means that you have nothing better to do with your income.
Refinancing your student loans from 4% to 2% is a fantastic idea, however (which is what the UK is doing)