Proof of work - lottery approach I'm sure I am missing something here. But not sure what ... 

Proof of work - after all is said and done, when a hash is found (about 10 min), it is said to be like someone hitting the lottery. Anyone that wins the lottery should be able to prove that a cost was incurred. So there are two things:


Include a cost to finding the hash.
The winner is considered "random" because finding the hash is an iterative guessing process.


The result is a huge energy burn and the result "mimics" a random winner. 

QUESTION: Can the bitcoin network randomly select a subset of the existing nodes, for each block, who then compete to solve the hash? For example, if there are 100,000 nodes, the network randomly selects 10% or 10,000 nodes who are allowed to compete to find the hash. Or take it to a smaller number like 1% or 1,000 nodes, etc.

The burning question is: Why can we not mimic the proof of work process and save a ton of energy along the way? 

Why not have all the nodes that wish to enter each block lottery pay a small fee. They all currently incur electricity cost so from a financial standpoint, does it matter whether they pay a small lottery fee or pay a small fee to the electric company? 

EXAMPLE: for the next block, here is the process:


All 100,000 nodes pay (arbitrary number for now to make point) 1000
satoshis.
The bitcoin network randomly (maybe this is tricky part)
selects a subset percentage. Say 10% or 10,000 nodes. These are the nodes that compete for the next block entry.


BENEFITS: 


There is still a cost incurred by each participant without the energy usage.
The energy usage is now a fraction of the total network participation for the iterative hash search. 
It can allow more node participation by less powerful computers. A laptop will still be at a disadvantage, but is now competing against 9,999 other computers instead of the hashing power of the entire network. 


To make sense of this is the fact that the end result of the current process is a "random" winner that can prove a cost has been incurred. 

Can the lottery fee approach achieve the same goal without sacrificing security?
There's a lot going, but I'll do my best.


  QUESTION: Can the bitcoin network randomly select a subset of the existing nodes, for each block, who then compete to solve the hash?


Selecting a subset of existing nodes isn't easy as it seems. The vast majority of bitcoin nodes aren't reachable at all, in fact. That's why every website that counts those are different from each other. (The best probably being Luke Dashjr's one).

So you might: "Well, you could make it so that in order to participate you have to public announce your node". But that opens up (from the top of my head) a big attack vector: DDoS's. If every node participating in the consensus is known and reachable then you could hit them once they are selected and cause a network failure.

But that's not even the most important part: This approach would be extremely easy to gamble. Any person could run thousands or millions of nodes (Amazon Web Service has 1 million business customers) and change the odds to their favor.

Proof-of-Work is not a consensus algorithm, but rather an Anti-Sybil protocol. Sybil being the attack that I described in the last paragraph.


  Can the lottery fee approach achieve the same goal without sacrificing
  security?


Absolutely and 100% no, if you take "without sacrificing security" serious. And (1) I'm not sure if it can be done at all or (2) if it would work, but regardless: no. 


  The result is a huge energy burn


This is a little off-topic but I feel that might be worth commenting. Bitcoin energy usage is far from "deadly" or something like that. Most researches point to +70% use of renewable energy in the industry, highest than any other.