That's such an dismissive and ignorant thing to say to someone in these circumstances. This has nothing whatsoever to do with martyrdom, and just the term being sent in her direction suggests she has something to do with this. Her company has a duty of care, and they are ignoring that duty over profit.
Having spent my 20s and early 30s been in hospitality management she is very clearly burnt out. I've seen it happen dozens of times, and when you get there, you get there with ZERO ability to find a way out of it alone! If your personal and financial circumstances don't leave you feeling powerless, mental health symptoms will.
OP - please try and understand that your colleagues have choices also. I know you know this, but I will remind you that ultimately, the responsibility for making sure shifts are covered lies with the person doing to rota. If your colleague can't stay on, she has a right not to and a choice, and if she doesn't, then above her will have to, until eventually the person responsible for doing the rota is covering that shift and being made to face what happens when you treat people like machines. I'm saying this because you need to ring in sick, as soon as you can, and sign off for at least two weeks. You have a responsibility to your family to do this. Your not going to be helping them with anything in hospital, or in crisis with your mental health.
Use the first week to recover, and the second week at your laptop to crunch numbers, reduce costs and plan what you want to do moving forward. And decide wether you can face going back. I have been exactly where you are now, and having successfully changed career, please please message me for a chat if it would help :)
Oh and...WTD is averaged out over a 17 week period. Its a common misconception that employers are breaking the law not allowing 11 hours between shifts or helping you rack up 71 hour weeks! Finishing at 2am to be back in at 9am for delivery was my norm most weekends for a long time! Along with going 10 days without a day off, only to sleep the entirety of anything i had.
Now in a Monday to Friday, 9 till 5 position in the public sector...I now suffer with crippling performance related anxiety, due to not feeling pain and exhaustion at the end of every day, and not spending my time off having anxiety attacks about mistakes I've made because I wasn't able to think straight.
I really hope you find the strength to make that call and soon.