A leader is supposed to do many things. One of all those things is "setting the tone". If you as a leader empower and favor a "bragging" and "bashing" behavior that is the behavior that will spread among your employees and will be part of your environment. There's a big risk that people will reflect the people that you as a leader hold high.
I'm a firm believer of direct and open communication. And I do not hesitate in taking a discussion and to deliver my opinions. And I do not mind people doing the same. What I don't like is people continuously bragging and bashing. Creating a witch-hunt atmosphere. How do you think that will be looked upon from outsiders or newcomers to your team? The risk is high that you will scare new members of your team from expressing their opinions. Scared from becoming a target of your loudly expressed criticism.
Even the smallest things...
Even the smallest things are part of "setting the tone". One example of something that we all see and are annoyed about, but often fail completely at, is "The messy kitchen". It's really simple to fix. All you need is some "common sense". But why should you care?
If I as a customer were at your office for a business meeting and would see a messy kitchen, I would take that as signs of bad leadership in the office.
How would I as customer know that this behavior only affects the kitchen? How about the rest of the work? Are corners cut there as well?
You are more important...
As a leader you might believe that your work is more important than others and therefore you don't have to mind chores. Of course your work is important. Maybe even more important. But that is the work of being an awesome leader that ensures that the company delivers as it should in an effective manner with low friction. That clear goals and strategies exists. That your employees are provided with all the necessary tools needed to solve their work. But part of being a leader is also to have a good manner and to show respect and interest in others. I once had the pleasure of working for a manager who every day walked around and said good morning to all the employees and consultants. His door was always open and he listened and paid attention to YOU, not just your work. He took part in every day chores. We had a great atmosphere and if he would call me today and ask for help, I wouldn't hesitate for a second to help him.