Why is POW/POS absolutely necessary to run public distributed ledger?
You can't do this due to the Sybil Problem. There is no way of proving a node is an individual participant in the system, you can't prove that a node existed at any point in time (or didn't), and there's no source of random numbers at all in the system which can't be influenced or altered. 

Proof of Work is a side step to having potentially clones participants in a system vote on validity, it replaces this is with a system in which members of the network verifiably destroy real world resources in order to create additions to the ledger. By adding a real world cost to producing alternate histories, it becomes infeasible for people to modify the history of the system and present it as the truth to anybody else. When faced with multiple choices, participants have a tie breaker by simply comparing the relative amount of resources destroyed and picking the most destructive one. 

In your described model:


How do you know that the network is really of this size?
How do you prove to a third party joining the network that you are not all the participants in the network pretending to be other people?
What is to prevent any node in the network signing many alternate histories in which people do and do not get paid, and using this to trick people out of money?


The proposed system breaks down as soon as you introduce malice, which is something cryptocurrencies must be design to defend against. Millions of fake nodes, lying, cheating, and denial of service attacks are all things the system must expect to encounter.


  at least from what I understand, that is what some POS coins do


Proof of Stake cryptocurrencies are usually completely centralized, due to the behavior you're describing here. Faced with the knowledge that PoS is not a method of distributed consensus they use centralized control in an attempt to ensure that the chain is coherent.