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.