You're basically asking a friend to be an untrained therapist for free. Even the most caring and generous of people know in their hearts that they can't take on that kind of responsibility and still be able to manage their own life. It's not fair to ask - most people will feel guilty about saying no to someone asking for their help, even knowing it's help they cannot feasibly give.
It's truly a full-time job. I don't mind paying my therapist, because while it could be interpreted as paying someone to care, I see it as paying for their expertise in helping me get better. The right therapist will care about you regardless of payment; it's both a professional and a profoundly personal relationship. It's why many therapists, particularly ones treating personality disorders, have to go into therapy themselves to stay grounded.
I struggle with depression and anxiety pretty severely at times, and while my boyfriend is tremendously supportive to an almost saintly degree, he absolutely insists on me having professional help. There is only so much he can do and asking more of him would be counterproductive for both of us. I wouldn't get better and he'd be taking on a problem that is ultimately not his.