Why is POW/POS absolutely necessary to run public distributed ledger? my first question here, so please be patient with me. Thank you in advance.

Q: Why exactly are POW/POS mechanisms needed to secure the blockchain (in bitcoin as well as in other alts)? + what is terribly wrong with my example below and why ? (I feel like I am missing something very important here). Thank you very much.

I mean: 
- why the block verifying and signing party cannot be just randomly selected node, that would every X minutes put new transactions into the block, calculate hash of block and sign it by its public key, broadcast it to other nodes for inclusion in the blockchain and collect some the reward ? (provided nodes on the network are able to reach some kind of consensus on selected random node - I know it is not easy task - but let's assume it can be done for a moment)

I mean, possible attacks on the network are still there, but it is same as with POW/POS, but:


if node is selected 'truly randomly' (I know, I know), the attacker would need to have just pure luck if he wants to fork the chain with his crafted transactions, because he does not know when he will have the privilege to sign the block... 
Or he would need to increase its share of nodes in the network significantly (same with 51% attack in POW/POS)


Above can be (?) mitigated to some degree with including check points in every N blocks in the best chain and having the rule of including e.g. hash of N-x block into recent transactions. (at least from what I understand, that is what some POS coins do)   

Why there needs to be POW/POS? (electricity burning/(locked) coins in wallet staking)

Thank you very much for putting my thought process back on track.
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.