Proof of work – lottery approach

I'm sure I'm missing something here. But I do not know what …

Proof of work – in the end, when a hash is found (about 10 minutes), it's like being hit on the lottery. Anyone who wins the lottery should be able to prove that a cost has been incurred. So, there are two things:

  1. Include a cost to find the hash.
  2. The winner is considered "random" because the search for the hash is an iterative riddle process.

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

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

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

Why not ask all nodes who wish to enter each block lottery to pay a small fee. At the moment, they all generate electricity costs, so from a financial point of view, is it important to know whether they are paying small lottery fees or a small fee to the company? 39; electricity?

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

  1. Every 100,000 nodes pay (arbitrary number at the moment)
    satoshis.
  2. The network of random bitcoins (maybe that's a tricky part)
    selects a subset percentage. Say 10% or 10,000 nodes. These are the nodes that compete for the next block entry.

ADVANTAGES:

  1. There is still a cost for each participant without energy consumption.
  2. Energy consumption now accounts for a fraction of the network's total participation in iterative hashing.
  3. This may allow for greater node participation by less powerful computers. A laptop will still be at a disadvantage, but is now competing with 9,999 other computers instead of the hash power of the entire network.

To make sense of this, the final process must result in a "random" winner who can prove that a cost has been incurred.

Can the lottery fees method achieve the same goal without sacrificing security?