How do I cope with a good mission done badly?

You believe in what this place is trying to do. That part is real. What tears at you is how it is being done, the sloppiness, the wrong turns, the good purpose undercut by poor execution. You are watching something you care about be handled in a way that undermines it, and the gap is maddening. The frustration you feel is a sign of something good, that you still care. Indifference would be easier and worse. But hold two things at once, the mission is worth believing in, and the execution is failing it. Those are separate, and confusing them will either make you cynical about the whole thing or blind to the real problems. Keep them apart. Then ask where your own reach actually extends. You may not be able to fix how the whole organisation runs, but there is almost always some corner, your team, your project, your piece, where you can make the execution match the mission. Do it well there. Fix what is in your hands, and let that be the answer to the frustration rather than swallowing the whole failing machine. Excellence in your own corner is not nothing. It is often how a poorly run good thing slowly gets better.