I'd certainly speak to your skip about this, but I wouldn't bother being overly diplomatic.
I'd focus on how the specific behaviour is impacting your work, and stick to facts, as you have done in your question.
I doubt that you'll be able to force a change in your manager's behaviour alone, but there may be multiple people that also feel that way. Hopefully they are able to approach the skip.
There is no need to worry about making the right choice for your manager. They have been at the company for 15 years. They are part of the furniture.
You should give you skip the opportunity to fix things, and before you become so frustrated that you do something stupid. The time to act is now.