What you are describing sounds like a typical sysadmin role, not SRE.
There are several principles in SRE which are specifically designed to avoid the issues you mentioned and to keep software engineering teams invested in reliability:
1. SLO is defined by the business, which describes reliability requirements for the system;
2. If the system is less reliable than SLO and runs out of error budget, feature development freezes, directing engineering resources to reliability improvements (this prevents the "last to be found" issue).
3. There is 50% cap on ops work that SRE do; everything over 50% goes to the feature development team. This prevents the "working your ass of to keep the system running" issue.
There are several principles in SRE which are specifically designed to avoid the issues you mentioned and to keep software engineering teams invested in reliability:
1. SLO is defined by the business, which describes reliability requirements for the system;
2. If the system is less reliable than SLO and runs out of error budget, feature development freezes, directing engineering resources to reliability improvements (this prevents the "last to be found" issue).
3. There is 50% cap on ops work that SRE do; everything over 50% goes to the feature development team. This prevents the "working your ass of to keep the system running" issue.
All of this is well described in the "Keys to SRE" talk by the guy who actually invented SRE: https://www.usenix.org/conference/srecon14/technical-session...