If you remove the requirement that a miner/node needs to "work" in order to find a block, it will become trivial for a single entity to create an ~unlimited number of "miners" from a small number of computers.
A very large number of "VM" miners but not unlimited - this kind of proof is actually designed to push people to develop efficient VM and container images just for sleeping - but like ASICs there's probably a performance ceiling to how many you can run at a time. In fact it could be the long-term heir to ACISs, since will cause CPU cores and threads to be hoarded instead of GPUs, FPGAs, and energy-eating ASIC miners, which are rendered useless for a proof of time or proof of idle model.
Regarding your specific proposal, network connections among all nodes are not going to have a small enough latency for it to work. Also, it is not uncommon for different nodes to have a slightly different times for a variety of reasons. Bitcoin's implementation of PoW, for example, allows for nodes to have their time be by up to 2 hours.
My parameters for sleep time and timeout are going to have to be tweaked to accommodate this.
With PoW, if a node is sent two copies of the blockchain, it is trivial for the node to determine which version is valid because it can trivially calculate which blockchain has the greater amount of total work.
I thought PoS also adhered to the "longest chain" principle?