From what I understand, we need mining to validate the block, we need the proof of work to limit the number of miners can add a block, and we need the block interval to give time to the new block to spread across the chain.
And usually it's said mining is like a lottery, and hence the alternative idea:
Instead of proof of work, we choose randomly some of the miners registered on the network, only that miner can validate and add the next block, and it has let's say 2min to do it, if not another miner is chosen randomly. In this way we don't need to waste a lot of energy to solve the hash problem.
What is the problem with this solution?
(I know it has a problem, but I can't see it and I think knowing this will help me to understand better why we need the proof of work)