At what point does a person do hospice care?

WHEN A CHRONIC. Health condition has rendered them unable to care for themselves and are terminal, expected to live 6 mos or less.
Sooner, not later. Most physicians are reluctant to refer to hospice because most think its only for the last moments of life. Hospice is more beneficial when the hospice team can build report and get to know the patient well. It will be easier to manage symptoms as they arise early on to improve quality of life rather than dealing in a crisis mode right at the end.
End of life care. Hospice is meant for those who seemingly at the end of life, although that end can still be many months away. The emphasis of hospice care is not trying to save the life, but to give comfort to the individual as their death approaches. The timing of death is variable - days, weeks, months.
Also... Patients can choose hospice care even if they are fully functional and can care for themselves. The only limiting criteria is their diagnosis of 6 months or less and their focus on comfort care. I've known people to work and still travel on hospice care. The focus is quality of life and the hospice team does everything they can to help accomplish that.