IAPT is designed to treat common mental health problems (you can find the list online: depression, GAD, OCD, PTSD, panic disorder, several others), however having a diagnosis of an SMI doesn't preclude someone from IAPT intervention as long as their presenting problem and symptoms is one of the disorders they treat, rather than symptoms of their SMI. Suicidal ideation isn't a barrier, however the presence of plans/intent does mean someone isn't suitable for IAPT.
If someone's presenting problem is their SMI or personality disorder they fall under secondary care services, or primary care hubs which have come into force in some areas of the country.
Lots of people claiming PIP and stating they're unable to work are saying it's due to depression and anxiety, which does fall under IAPT. People with SMI whose symptoms are still very much florid will need treatment for something IAPT doesn't offer.
Risk of self harm isn't a concern, if it is not "dangerous", lots of people self harm, it's a part of so many mental health problems and a way a lot of people manage distress.
It's a shame there is such misconception out there about what NHS mental health services do and what's available.