Isn't the necessity of skip meetings itself a pathological sign? The only thing you legitimately need to talk to your manager's manager about is things your manager can't handle -- i.e. mostly problems with and/or praise for your manager. Everything else should go through your manager, or what are they there for?
Depends on the org and the situation. Sometimes the guy a level up has a broader perspective.
Like anything, ymmv. If it’s setup as a way to rat out your boss in a deep org, that’s stupid. If it’s a way for the director to establish relationships with some of the ICs, that can be productive.
When communicating through a middleman you always lose information or it gets distorted. There is definitely value in talking to upper levels directly.