in a distributed system, conflicts are resolved by majority vote, and voting has to be made expensive to prevent stuffing the ballot-box. Proof-of-work is one way to make it expensive. Proof-of-stake is another.
I have read the thread, and there was a convincing differentiation of shares vs. coins.
With shares, you vote without working, according to how much you have at stake from previous investment.
With coins, you must pay for every vote you want.
The first one is scarcely suitable to be a basis for a monetary system, because there is no anchor, no cost that "keeps people honest" like in gold standard vs. fiat standard.
Switching back to fiat after just 5-8 years of enjoying the renewed benefits of gold would be sad.
With shares you also have to pay to vote, since you have to pay to acquire shares!
I don't get why people are so prompt to make the analogy PoS=Fiat. In fiat people who vote (Central bankers) have divergent interest with fiat holders, they have no skin in the game. Whereas with PoS the interest of the decision makers are the same than the interest of holders. Skin in the game is the way to keep decisions in check (and that's why everybody right now is fine with miners making decisions: because like holders, they have skin in the game).
Also the argument of the value derived from cost has the flavor of the marxist labor theory of value. The value of the network doesn't comes from the cost to maintain it but from the services provided by the network (the more useful the services are, the more there is adoption, and the more the price increases).