It really depends where you work. My team (community mental health service in the NHS) has a high amount of sickness due to stress and a couple of chronic conditions. There were times I was the only B6 nurse for many weeks on end.
I myself went off for three months whilst pregnant as my hormones triggered my OCD and I was in practically daily hysterics and not coping at work. A colleague went off for 6 months with bereavement. Another colleague went off for 3 months following hip replacement, then came back, then had the other done and went off again. Another colleague goes off quite unpredictably due to chronic migraines.
I've been off for 2 episodes of stress. The trust only starts proceedings when you've had 3 sicknesses (pregnancy doesn't count). The only person I've known to have been made to retire on ill health is someone who was out of work for nearly 3 years on and off due to chronic lung problems. I've never once been hassled whilst off sick with mental illness. I just have regular supportive contact with my manager.
OP is stressed and not coping. Many people would be the same with a kid in hospital. Stress is a valid reason for a sick note. I genuinely think people would not be coming down on OP if she needed time off for something physical. Also, stress IS physical as well as mental.
OP probably wouldn't be suffering if their child was well, but they aren't. Sometimes we become ill due to stressors in our life e.g bereavement, carer stress, etc. That's like saying 'well OP wouldn't be suffering if they didn't have cancer' or 'well OP wouldn't have developed depression if her dad hadn't died, so the root cause is compassionate reasons'. I wouldn't have gone off sick if I wasn't pregnant either, but it's a moot point, because I was.
I truly truly don't think any reasonable employer is going to question someone going off with stress with a sick child.