Actually a lot of cities have city run mental health facilities that are for low income people, although I wouldn't know if any are near you.
And they are so full of people the ones that need it can't get in.
Private mental health facilities also charge upwards of three thousand dollars per day or more (3k is just the highest around here), people can involuntarily land a 96 hour hold for evaluation, most of them are actually a danger to the patients trapped within rather than some form of treatment, and to ensure insurance payouts everything is charted in such as way to make it seem like you needed to be there which has lasting repercussions on further treatment.
Also there is no such thing as free. Medicaid which pays part of their bills for them if they are too poor, hospitals with a county contract are also given tax revenue for compensation for various emergency services, and there is even additional state/national funding if they also happen to have no competitors since they become the only source of true emergency healthcare in the area. Every productive member of the USA pays into funding this, some more than others.
This may blow your mind but healthcare isn't free and health insurance isn't insurance.