Is there an outpatient clinic or a mobile ACT team in your area?

Most likely they are not taking you on, due to the fact they feel you need a higher level of care (based on your own reporting of symptoms ie cuuting, active suicidal ideation), not because they don't want to help/don't care/are a joke.

A private based practioner will not take someone on who is not seeing a psychiatrist (or will not see a psychiatrist) and/or on any meds and experiencing the symptoms you are describing because of the high risk you present.

It's not about rejection - it's about approriate levels of care. Therapy as an only modality of treatment is not efficacious or appropriate treatment for someone in crisis mode, as you yourself describe. Please consider trying a clinic for starters, instead of private practice therapists.