The PoW job is also to maintain the Block Time at 10min. We can achieve that by just making everyone wait for that 10min minus delay/network broadcasting time etc.
Say round 1 only takes 2 minutes to get 100 miners. Then you just wait 8 minutes to select the "winner". That's a 10 minute block time. So it saves 8 minutes of hashing. Less global warming.
You would need a way to prevent the knowledge of the winner for 8 minutes so that nobody can start on the next block before the 10 minutes have past.