Based on the Dodo Bird Conjecture*, I don't even think there's a reason to think that AI would do any worse than human therapists. It might even be better because the distressed person might hold less back from a soulless machine than they would for a flesh and blood person. Not that this is rational, because everything they tell an AI therapist can be logged, saved forever and combed through.
I think that ultimately the word we should use for this is "lobbying." If AI can't be considered therapy, that means that a bunch of therapists, no more effective than Sunday school teachers, working from extremely dubious frameworks** will not have to compete with it for insurance dollars or government cash. Since that cash is a fixed demand (or really a falling one), the result is that far fewer people will get any mental illness treatment at all. In Chicago, virtually all of the city mental health services were closed by Rahm Emmanuel. I watched a man move into the doorway of an abandoned building across from the local mental health center within weeks after it had been closed down and leased to an "tech incubator." I wondered if he had been a patient there. Eventually, after a few months, he was gone.
So if I could ask this question again, I'd ask: "What if it works 80%-120% as well as a therapist but is 100 or 1000 times cheaper?" My tentative answer would be that it would be suppressed by lobbyists employed by some private equity rollup that has already or will soon have turned 80% of therapists into even lower-paid gig workers. The place you would expect this to happen first was Illinois, because it is famously one of the most corruptly governed states in the country.***
Our current governor, absolutely terrible but at the same time the best we've had in a long while, tried to buy Obama's Senate seat from a former Illinois governor turned goofy national cultural figure and Trump ass-kisser in a ploy to stay out of prison (which ultimately delivered.) You can probably listen to the recordings now, unless they've been suppressed. I had a recording somewhere years ago, because I worked in a state agency under Blagojevich and followed everything in realtime (including pulling his name off of the state websites I managed the moment he was impeached. We were all gathered around the television in a conference room.)
edit: feel like I have to add that this comment was written my me, not AI. Maybe I'm flattering myself to think anybody would make the mistake.
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[*] Westra, H. A. (2022). The implications of the Dodo bird verdict for training in psychotherapy: prioritizing process observation. Psychotherapy Research, 33(4), 527–529. https://doi.org/10.1080/10503307.2022.2141588
[**] At least Freud is almost completely dead, although his legacy blackens world culture.
[***] Probably the horrific next step is that the rollup lays off all the therapists and has them replaced with an AI they own, after lobbying against the thing that they previously lobbied for. Maybe they sell themselves to OpenAI or Anthropic or whoever, and let them handle that phase.