Wait what? Why are you putting yourself and your team in a position where your reports can screw you over? And if one or two people are doing whatever that is, why aren’t you firing them or managing them out?
I don't have magician or telepathy skills, I don't immediately see the problems. Even on 1:1s I can be persuaded and believe lies.
And well, for reporting I kinda agree, we don't do that many reporting as others. I would say other managers need advanced tracking and reporting to automate their own reporting to higher management.