SREs are not on-call, dashboard, and deploy monkeys. They are software engineers who focus on reliability tasks not product tasks. An ideal structure requires all engineers to perform basic operational tasks including on-call, deployments, monitoring, etc. I think this is critically important as it helps to avoid class/job stratification between reliability and software engineers and makes software engineers more directly accountable for product quality.
All engineers do not like doing all things. Some engineers like being generalists. Some like specializing. Some like writing code. Some like debugging. Some like UI. Some like systems. A growing engineering organization that supports specialists has to grapple with the fact that employee happiness sometimes involves working on certain types of problems and not others.