POS=no production cost=no value
If I can produce a coin without any cost, why should I pay any thing valuable to exchange it??? I will just go ahead to mine it
Fiat money can have value without production cost, because there is a law to force its circulation, and the cost to make that law can be a war, which costs millions of times more energy than POW mining
You make a common mistake here.
PoS has a production cost. To invent a working PoS system, you need to do brainwork. To generate a PoS block, you need to do CPU cycles (and you need to have stake of course).
But that doesn't matter, because production cost != value