Now that I think about it, military spending isn't entirely "money sink". It stimulates the economy indirectly. Via the state owned companies the money flows to private contractors and workers, both utilizing it again, and now elsewhere in the economy.
Also rearmament was also a popular move, great many in Germany would wanted to see their country regain her rightful place as a world power as it was previously. And for that at the time proper army was needed, especially if they wanted to be taken seriously by other powers, particularly at the conference table over maps revisioning borders and such they oh so longed for.
And again I'm compelled to write about the volume of the rearmament. It seems a lot because it they had to build from almost zero. But it's extent worth only something if we compare it to other nations.
When Germany attacked France, they stood about on equal grounds, tho France had more tanks and Germany more airplanes.
When Germany stroke the Soviet Union... the latter had way, way more technical equipment, both in tanks and aircrafts they had overwhelming numbers... and this is after the German military industry started really booming. And the Soviet Union paid the price dearly, for all those equipment to have they sacrificed not just the standard of living but actually the lives of millions.