Depending on the team, SREs can absolutely involved with "building the system", especially the klaxon ;) Examples include designing and implementating metrics used to make make decisions in business logic and or exposed to customers/users, writing routing components like mixers and proxies, developing data pipelines, etc. At Google many SRE teams build and run entire multi-tenant systems with no pure SWEs involved at all.
Healthy SRE teams should be spending 20% of their time on operations. On my team its actually the devs who do most of the operations work. They take the pager during business hours and we route most maintenance tickets to them.