You're right that Y11 is not the ideal time to be starting home ed, but at the same time, you don't have the option of choosing to go back in time and do it a few years earlier. The question now is, what is the least bad option?
I would say it really depends how bad the effect is which school is having on your daughter's mental health. Mental health trumps everything. Education is important, but it's never too late to get a good education. It's true that GCSEs are easier to get via school, this year, if that's an option, but the toll that takes on your daughter may be too high.
It may be too late already to jump ship and sit exams as a private candidate. Some GCSEs are not accessible to private candidates, and so home ed kids often do IGCSEs instead. It's a completely equivalent qualification, but the specification is different and in some subjects that means significant extra work. You'd need to find one or more exam centres which take private candidates; not all centres will offer all the subjects you want. Some will be fully booked by now, and the registration deadline for others will have passed.
If school is intolerable, it may be worth taking your daughter out and then stopping to take a breath while she recovers, and she takes stock of her overall situation. Exams at 16 are not compulsory. Schools act as if they were, because there's a lot of pressure on schools to push kids through as many exams as possible by that age in order to "prove" the school has educated its children well. But outside the school system, it's a different picture. You can do exams at any age. You can do as many or as few as you want. You can spread them over several years to reduce stress. You can wait until after 16 and do some GCSEs at college for free. All colleges will have some catch-up options, though they won't offer the same range of subjects that schools do.
Most home ed kids who have the capacity for GCSEs do at least a handful at some stage. Some don't. My eldest did one GCSE at 19 and another at 20 and is now doing very well at uni in an arts subject.
So your daughter could leave school now and focus her learning on something other than achieving a set of exam results in the next six months. That is perfectly legal. What does she love doing, academic or otherwise? What makes her come alive? That is often the key to restoring mental health.