In our weekly meetings, recurring problems were identified and fixes implemented. No call was considered completed and closed until all relevant documents had been updated as appropriate. At the yearly review the quality of your documentation was as important as your time to respond, time to fix. That is how mission critical on-call work should be handled.
I wrote a longer guide about this here: https://onlineornot.com/incident-management/on-call/improvin...
I think there should be a nice light weight tool which should give a clear summary and tracking mechanism which make this a quicker tasks. Even just to tag the runbooks which are not updated. All those notes get lost in documentations and never referred back.