1. You need some way to enforce the rule which prevents minters from minting every block. In PoW/PoS this would be:
Indeed - the approach that CIYAM uses is known as a "hash chain" (which functions like a PRNG meaning that the next best choice of minter is random).
2. What benefit is there to only rewarding every X blocks, instead of rewarding reward/X every block?
The question is whether there needs to be any "reward" at all (assuming that you don't require increasing costs of hardware and electricity in order to run a full node).