Manager won't manage and sh!t flows downwards

Hope to keep this succinct while still giving enough information to get your insights.

tldr: 50% manager refusing to manage, just bounces everything up to their manager who does nothing

Public sector organisation, I'm 5th grade from the top, manager is 4th. We're a small, under-resourced (verifiably and have been for 4+ years). Manager is over another project which on paper takes 50% of their time but realistically it's more (arguably more interesting, less contentious work, closer to their background, they resent being given our 50% when they felt they were hired for the other role only).

Additional staff have been semi-promised for 12 months but no sign. Manager very poor at prioritising work and doesn't understand that by delegating work to us (that other managers would take on themselves) it makes it even more difficult to do the statutorily mandated job we have to do. Atmosphere is getting toxic and everyone is worn out. In fairness I'm probably the least stressed so far but the more it goes on the more the whole situation is bringing me down. I'd normally be pretty resilient.

Raise these issues with manager and they say 'I've passed this on, if you want a conversation with X (their manager) feel free, I don't know what's going on etc'

I feel if my manager took some responsibility, properly and strategically took decisions about what we were doing and who was doing what instead of leaving the team (all the same grade) muddle through things would improve. This, conveyed politely, isn't landing.

What would you do? Thanks for reading.