But interestingly, we solved these problem back then using an internal tool. Here is how the internal tool solved the problems -
1. It had integration with all internal tools like task management, alert management, monitoring systems and pagerduty. It offered 1 central dashboard where it all came together
2. Each team in pagerduty can see details of all alerts that happened in given shift/rotation. So anyone can go there anytime to see what alerts are fired, when and to whom.
3. Each alert had option to mark with various tags like noisy, non-actionable, etc. Additionally a note and follow up task links can be added.
The tool solves some of the problems you mentioned properly. e.g. You don't need to write a summary document. It all gets captured there and can be easily viewed in the handoff meetings. With tags, you can easily find bad alerts or alerts with outdated runbooks. Its easy to hold team-members accountable if they are not following process/best practices.
Oncall is a heavy process but IMO with right tooling, a lot of problems can be solved properly.
PS - I didn't create the tool but I used it extensively to get my team's oncall under control.