So you are saying the direct cause of lower unemployment was a slashed federal budget?
Personally I would look more on the logical premise of exports during that period increasing exponentially. You're talking about an era where manufacturing was literally everything. This is one of the periods in which the US had the biggest economical advantage arguably in the world. Just a few things to pay attention to:
-WWI left a massive opportunity for exportation to Europe (which had plenty of capital)
-Cheap, abundant labor because of domestic layoffs from WWI (reason for high unemployment)
-The introduction of credit as a means of purchase. Credit was introduced in these years maybe meaning aggregate demand had a positive effect, or it was a contributing factor to the bubble we know of prior to the Great Depression
Personally, I think the unemployment was decreased because of such a drastic rise in exports to fill the gap where other countries had to rebuild, the introduction of credit as a means to drive demand (short run, ie your example of when unemployment dropped then sharply rised thereafter), an overconfidence in investment throughout the US, and probably some inefficient government programs were slashed correctly (although this is probably the most minimal impactor).
You should not try to force every argument you make to back up the Austrian Economics Business Model. There are tons of factors that come into play when you are talking about an economy. Employment/government spending are NOT direct cause and effect.
