Don't feel bad about 'blowing your trumpet' or complaining; that's exactly the time to do it!
My manager and I skip our 1:1s more often than we have them, and when we do, it's usually just a hang-out session.
We meet once every two weeks, and most of the time is just talking about random things we found neat.
However, there are times when another team is being difficult and I need help 'navigating the business'. These meetings are super valuable for that.
I can provide more in-depth information on the situation as a whole, and we can work on a plan - without needlessly involving the rest of our team.