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.