So since the place is clearly clueless, you need to decide what your going to do. When you talk to people about this place in the future, they are going to ask you "why you stayed" and your answer is going to reflect on your character and your outlook.
You can try various strategies to fix things, some might help, some might not. But you can practice you skill in identifying roadblocks.
You can switch into anthropologist mode and start studying the people who are dysfunctional. Find how how complete idiots maintain a lock on their power base, what techniques do they use if it is clearly not skill in their job.
You can practice organizational dynamics, try to figure out who the 'players' are and who the 'pawns' are. It's a sort of morbid curiosity and of course once the players notice you are looking at them you will have to be on your toes and bob and weave lest they corner you on the board. (You don't want to end up fired with everyone in the organization thinking you're the reason the world is screwed up, and a good player will try to make that happen to protect their turf)
I wish there was a way to quit and on the way out mark the entrance with ultraviolet sensitive ink or something so that others could know what goes on there and stay away.
Since you are asking these questions though, you are in good shape.
While this is extremely interesting, it's not something that can be learned in a reasonable timeframe, and may never be learned if you don't have the right approach or luck into the relevant facts.
For example, one company I know of, the CEO was in charge for 8 years, while the company constantly lost money. They struggled to come up with new products, flailing around. They only stayed afloat through patent lawsuits, without which they would have been long out of business.
During this entire time, the CEO made mid-six figure salary. I use to remark, "If I understood why that company was still in business, I'd be rich".
Why did the CEO stay in charge? The answer is very boring. 2 of the 3 board members had already checked out, writing off the entire investment, and the third was personal friends with the CEO.
The answer is not always interesting, and there's not always as much to learn as you'd hope.