Authority: I teach Human Resources to MBA students.
There’s no specific actions I can suggest, but I can offer some food-for-thought and general comments.
First question is who will suffer from the results of their incompetence? If you’re part of a large organization, so that sooner or later their errors will catch up to them, then to a certain extent you can disassociate yourself from it. Do your best to catch and correct what you can, but shrug your shoulders at the rest. It’s not your responsibility to make incompetent superiors look good. Neither is it your responsibility to help sink them, of course.
If you’re not part of a large organization, so that there is risk that the customers and business will suffer badly, then you probably need to be more dililgent in trying to catch errors.
When you do find goof-ups, do your best to be non-critical (that is, don’t make value judgements.) Report factually, avoid “loaded words” that might give offense. WRONG: “You stupidly couldn’t even add the numbers correctly.” BETTER: “There seems to have been a typo here.” Think to yourself: we’re all human and we all make mistakes (some people make more than others), and so everyone needs back-up to help catching and correct mistakes.
I don’t think I’d use emails for that, unless there was no choice. Emails can be misread in terms of tone. You want your tone to be friendly, helpful, cheerful, matter-of-fact … not critical, not “gotcha again”, not scolding. That’s best done face-to-face… although you might document in an email later, “Just a reminder that …”
BTW, I would be sure to start documenting. It might just be a little diary you keep for yourself, it might be emails recounting meetings, whatever. But you might want to Cover Yer Bum (CYB) by having documentation. Memory plays strange tricks if this explodes months from now.
Now, having said all that, I suggest that you need to think carefully of what the future is likely to be. Think realistically. What do you think will happen long term? And then you need to ask where you want to be or need to be.
For instance, if it looks like a trainwreck is inevitable, with the whole thing going belly-up, then get the hell out as fast as you can. Maybe an internal transfer, I don’t know what the place is like. OTOH, if you think the dumb clucks will be fired and you’ll come out of it OK, then ride along and CYB. Bottom line: analyze what you think will be the long-term outcome, and (given that outcome) what’s the best thing for YOU to do.
Good luck. It’s really tough when your boss isn’t doing his/her job. Leaves you kinda stuck in the middle.