This is precisely what's happened in our company. The project management head was not a servant leader and actively avoided doing real work while telling others what to do and promising to do things that never happened. Sadly, this person is still with our company at the time of this writing, but thankfully not anymore in a role where they can derail work.

What really struck a nerve in me was that I had to catch all the work this person wasn't doing, and I only realized it after I got burned out from all the extra work.

The bigger issue then is that you can't solve this because often, the person does a good job of hiding the bigger issue by deflecting to smaller, more pressing ones.