I don't know if this is the best approach, but I actually kind of like these teams, and like working with/leading them, because usually the description is not true. And careful rebuilding of trust can usually (but not always) demonstrate that.
Very few people are truly as ineffective as they might seem, and people are more impacted by situations they are in, or the process/role/authority they have, then anyone will care to admit.
> in many cases who have demonstrated repeatedly that they make lots of mistakes and get sloppy and will fully refuse to cooperate
Ok, sure. I agree, and have seen this happen too. But why? Very few people intentionally want to make mistakes. Very few people intentionally refuse to cooperate for no reason. Most people are not inherently bad.
Are they making mistakes because they don't know something? Are they making mistakes because they are used to working in a different way, or prefer a different way? (If so, why?) Are they making mistakes because they feel some sort of pressure (like a time crunch, budget crunch, some 'three strikes' hanging over their employment status, etc). Is "only motivated by a paycheck" even a bad thing? (People can still do good work, despite only being motivated by payment. That's how most other industries work already). Are they refusing to cooperate because they have an unresolved grievance (where they demoted, or docked pay? Are they working on something that is slow/bad, because they couldn't get buy-in to do it the proper way the first time?)
People can be kind of cloudy, we're not robots, and there's not necessarily one single right way to go about rebuilding trust and creating functionality out of teams. And yes, occasionally there really are some folks who need to specifically be dealt with. But I really bristle when whole groups of people get written off with labels like the above, because it feels fundamentally untrue for most people most of the time.
These are all the types of things that a good manager is supposed to be solving. A good manager can make a "highly ineffective team" effective, even if they might never have been before.