Even better, get out of the private sector (if you can). I'm very happy with what I'm contributing to the world at the BBC, and I'm paid 'well enough'.
The BBC might be an exception here but typically government projects always go to some BS consultancy that will take lots of money and deliver a shitty solution, or sometimes the problem or spec itself doesn't make sense but you're not allowed to talk to them and clarify/rethink the spec because reasons.
There are some notable exceptions aside from the BBC in the UK. The Government Digital Service is held in fairly high regard. I’m not sure about the states. There are no doubt examples at the BBC of what you describe, but mainly it’s pleasant smart people building useful things. There’s dysfunction, but it’s bearable.
