So, not all hospitals have inpatient psychiatric facilities, but most have emergency rooms. If they aren't equipped with an inpatient facility, obviously there's no place to admit them to and all they can do is either keep them in the ER for a few days for observation or send them to a facility that does have an inpatient psychiatric unit. Also, if they don't have any psychiatric units at all, I can see how their ER might lack a mental health provider who can adequately assess a person who comes in on reports that they reported suicidal ideation in a therapy session with their outpatient provider. However, any hospital with an inpatient psychiatric ward will usually have what's called a comprehensive psychiatric emergency program (CPEP), which is effectively a psychiatric emergency room (sometimes integrated into the actual ER, sometimes not), and they obviously will have psychiatrists to assess patients who come in during mental health crises.

My point was simply that outpatient therapists and psychiatrists are fully capable of assessing the seriousness of a patient's suicidal ideation and that a trip to the ER isn't always necessary. Patients should only be sent to the ER against their wills when the outpatient provider is not convinced that the patient won't harm themselves before their next session.

Sorry, I know that was wordy, but I hope I answered your question.