I'm thought experimenting on also having a README written by direct reports that the boss doesn't get to see (so people could be more honest). I'm guessing it would quickly leak to the boss anyway, or someone trying to score points with the boss would just whitewash it. Perhaps only the person who wrote something could edit it, so you get lots of perspectives?
Would be an interesting experiment tho.
Why not do both? "Hey folks, here's my README, I'd encourage you to write something similar and share it with everybody."
When it comes to team building, I think it's much more valuable to say "this is how we operate" instead of having everyone write down "this is how I operate". The former is a mutual guidebook for everyone to work together, while the latter feels like an edict that everyone else needs to work around you.
Check for any overtly abusive behavior that requires correction. Usually rare, but worth checking.
Provide basic coaching and suggestions for working together in a healthy manner.
And most importantly: Let them know that they're expected to figure out how to operate together on their own like professionals. Managers can't be mediators for every impedance mismatch between employees.
I'm certainly not writing anything about my personal life because that's personal and, quite frankly, none of your business. I have a tragic backstory that I don't want to share with coworkers. It's my job to overcome my past and be effective and productive at work, not everyone else's job to accommodate any unexpected or unprofessional behavior, etc.
There's a whole host of instruments that more mature companies use to uncover the different working styles and professional psychologies of employees. E.g. knowing that one colleague wants just the facts and isn't interested in the back-story whereas another wants lots of context can be very useful and ultimately help one become more effective professionally.
tl;dr, Manager README's are more for the manager than the direct report. It's a great exercise in introspection if kept private and not used as something codified for employees to follow.
1) It's manager-centric. Why would you give employees a guide to their manager when you could give them an "Employee README" that is focused on the new employee? When I join a company, I want to know what's expected of me so I can succeed. If the company delivers this in a form of a manager README then I'll still be missing the big picture goals of the company. Managers - Focus on your employees, not yourself.
2) Manager READMEs become an aspirational target. Describing yourself in text is challenging for the ego. I've seen too many manager READMEs that read like a picture-perfect version of what the manager aspires to be, rather than a description of how the person actually operates.
3) The self-promotion aspect. When managers post their README all over internal company chat, public Twitter, their LinkedIn profiles, Medium think pieces, and the company blog (like this article), the document becomes a brand-building exercise. This puts even more pressure on managers to only write down their most perfect, idealized vision for what they want to be. Or they spin their flaws as some sort of superpower with creative wording. I once read a public manager README that was nothing like that person's actual management style, but it read as if that person was a textbook-perfect manager.
My recommendation: Skip the manager README fad and focus on your employees. Write an "Employee README" onboarding guide for each of your employees, that you customize with individual expectations and information for each employee before they join the company. Keep it in a shared document that you can update over time to include an up-to-date job description, list of responsibilities, and key expectations for that employee. Don't try to make management all about the manager's personal brand.