Haven't worked at a startup yet with a sysadmin. They either have devops or SREs take care of infrastructure but those types tend to be cross functional so they're also handling infrastructure, operations, incident response, developer tooling, CI/CD environments, code performance issues (basically everything that's not application development).

One company I was at heavily used SaaS and AWS for everything so there weren't very many systems to manage. We'd just jam everything in Docker containers and use AWS ECS. Patching hosts was just rotating to a new AMI on AWS; patching applications was updating dependencies.