In my opinion "spending vs. savings" is not the best way to put it. It would be better phrased as "debt vs. savings", or maybe even "growth vs. stability". I don't think even an Austrian School economist would argue that spending doesn't drive an economy -- that's what an economy is -- but rather that spending driven by debt is problematic. The argument is that more debt in an economy creates inflation by definition and destabilizes the market. They will cite countries that have less debt (among businesses and individuals) tend to have more stable economies and higher exports. Of course, this is partially because they are selling to countries with higher debt, so I don't think it's as simple as either.

EDIT: Also important to point out that Austrian school economics are against manipulation of monetary policy. They argue that interest rates, credit and savings ought to be driven by market factors rather than the government. It's not so much "debt is bad", but rather "artificially driving up debt by manipulating monetary policy is bad".