The military has a potentially infinite demand to be exploited and all of its funding comes from taxpayers. This rearmament is essentially forcing the people, who would otherwise not pay for consumer goods, to buy military items, though they do not contribute to living standards (Tooze notes they did, in a certain way). Since Germany's factories couldn't find buyers, taxpayers were herded into being its clients. It's forced consumption with no direct increase in living standards.
But the "money sink" interpretation makes the whole recovery seem even more ridiculous. The Reich used an irregular accounting scheme (Mefo bills) to pay money that didn't exist (but was promised to be paid back with interest) to companies producing dead-end goods that weren't productively used and then taxes those companies to artificially boost exports. It sounds like a pyramid scheme, though all stimuli of this kind sound absurd, with Keynes even claiming that burying bottles filled with cash in abandoned mines and letting the private sector dig them up would be a net benefit to the economy.
Things start to make more sense if military spending is, instead of a bottomless pit, an investment to be paid back in the future through war.