> My main concern is that this job is slowly making me sick.
This needs addressing sooner rather than later, either by fixing the problem(s), managing to care less, or getting out of dodge. It sounds like you want option 3 but are having trouble finding somewhere to go to.
For fixing the problem: make sure that you definitely understand the problem. This sounds odd (surely you know what is going on in your own head?) but it is often difficult to see the rotten needle of wood amongst the treestack. Are there other problems with the job, and this is the final straw or the squeaky wheel?
> the slow and low quality outputs
> it may be that the only way to fix is to address the inputs
Yes, you may be looking too much at the symptoms of the problem - the slow and low quality outputs, and the only way to fix those is likely to be to address the inputs by taking time out to try to retrain the team for the working patterns needed. That will mean more delay short term, because that training/practice will impact on time available for work, but could pay dividends in the medium/long term. It is worth a shot pushing that option while you are still unable to find work elsewhere: if you can't find the job/team you want, then try to mould this job/team into something better! If either the team or the management above you resist this, then there might be little you can do, but it does seem to be the way forward for the company (if they can't accept that then the company may fail, and you'll no longer have the stay/go decision to make - if management resist then make it clear that you see this as the risk of inaction).
If you can't fix it, and can't find better work elsewhere, then you might have to think about a drastic move to something else entirely, at least temporarily. This may well entail a pay/benefits cut though, so not something you can do unless quite financially secure. Very drastic, but you only have one mental health, and it can be very difficult to repair if things go to far. Your mind isn't a server that you can just pave and restore from backups!
What other people say about “trying to care less” is correct, especially trying to stop thinking about it in your personal time, but if you are anything like me then not caring feels wrong so it is very hard to turn that off (or down much).
Reference: I was suffering burn-out a couple of years ago due to a mix of work issues, personal responsibilities/stresses (ill parents, nearly lost both of them, have essentially lost one due to dementia, a lost cat (lymphoma, poor lass), and so forth), and my own mental issues (I'm “more than a tad bipolar”). Dropping to a four-day week has helped a lot, giving me more time to sort my self out or just to relax, and I've been doing that for two years. It turns out though (this is my personal “wood hidden by the trees” thing) is that I don't cope well with remote working long term, being alone with my own thoughts and no other real company (the cats help if I'm working from home, but I can't talk to them about work matters!) is not good for me. Mid/late 2020 is when I really started going down hill (though I didn't quite see it at the time). Even now I'm in the office four days a week and so are some other people, most of the people I'm working more directly with usually aren't, so I'm still effectively remote. I was starting to get a proper handle on all this when AI hoved into view, and I really don't want that for a list of reasons to long to rant about here. At some point very soon the combination of not getting on with remote and not wanting to get on with AI doing my job are going to be too much and I'll go. I'm lucky though: mortgage paid, no partner nor kids, some savings - which means I can afford to take the cut needed to just go when I need to. Heck, I might end up waiting tables or being a hospital porter, there isn't a lot that I'm qualified for that isn't going to be both remote and increasingly steeped in AI… The thought that I can do that is actually helping with the current situation, reducing the feeling that I need to do that, so keeping muddling on, doing the job while I can and putting the money safely away, is more palatable, giving me more time to research future options (or see if the balance changes much when the big AI firms start trying to recoup all that investment (https://isaiprofitable.com/) and AI-driven work looks less inexpensive).