Elastic compute means you want to be able to treat compute hardware as fungible. Persistent local storage makes that a lot harder because the Cloud provider wants to hand out that compute to someone else after shutdown, so the local storage needs to be wiped.
So you either get ephemeral local SSDs (and have to handle rebuild on restart yourself) or network-attached SSDs with much higher reliability and persistence, but a fraction of the performance.
Active instances can be migrated, of course, with sufficient cleverness in the I/O stack.