I'd tend to agree that psychology can be unscientific at times. Certainly a lot in that field is derived from very abstract theories (e.g. psychodynamics) that are not as falsifiable as one might hope.
Yet, the comparison to astrology is also unfair. Scientific studies have repeatedly shown that many psychological treatments (e.g. CBT, DBT, interpersonal therapy) are effective. A lot of early medicine worked this way: we knew a treatment was useful before we understood the fundamental biology of how it worked (e.g., rudimentary vaccines came before germ theory).
You are making a false equivalence between clearly identifiable physical disease (e.g. shingles), and this questionable concept called "mental illness" or "depression", and this false equivalence is at the root of the problem.
This is one of the problems of how the scientific revolution has been carried out in western society. The scientific method which was so successful in application to certain problems (mainly in the physical realm), has been assumed to be just as effective in other areas as well.
Then we see false inferences being drawn and faulty conclusions being reached based on philosophically questionable assumptions being drawn - "hey, it looks like we are applying the scientific method, therefore we must be right!". Badly applied science is just as bad a non-science like astrology.
Guess what - if you lock a chimpanzee in a cage it will become "depressed". Maybe just let it out and return it to the jungle? That ain't science, it's just common sense.
If you run a bunch of people who are unhappy through a standardized psychology program (e.g. CBT), they get happier compared to the people who didn't get that program. That's just a fact.
Maybe that's not the scientific method by your definition, but I'm not sure it matters.
And yes, I get that a lot of mental health problems are caused by problems in society. To extend your analogy, we're not the zookeepers, we're other chimpanzees. We're a lot smarter than monkeys so we even perhaps have a shot of escaping. But it is absolutely reasonable to try to help each other make the most of our lot.
So who care's what you call it, therapy is a way to help people have more fulfilling lives. Medications can be too.
Again you run face first into the brick wall of questionable assumptions and get a bloody nose.
What's happiness? How do we measure it? How long does this feeling last? Was it measured again 10 and 20 and 30 years later? And what was the "control measure"? People who did not go through any program at all? Was it simply the opportunity to think through their life issues rather than the "CBT" itself the main driver to improve their mood? Did the study attempt to identify different causal factors? Did the participants actually make substantive changes to their lives or did they simply start to "feel better"? Did different people with different backgrounds respond differently to different kinds of treatment? (rhetorical question obviously).
Are you now starting to get a taste of the absurdity of psychology and psychiatry? Sure getting people into a room to simply start talking and thinking about their lives is a good thing. It probably helps, in different ways for different people. But again, this is not science, just common sense.
The attempt to "sciencify" and pathologize lived human experience is like a kind of mental corruption by the scientific establishment that refuses to believe anything is beyond it's scope. But again, just because you run what looks like a "scientific study" and write a fancy "scientific paper" does not validate the corrupt and invalid assumptions and beliefs it is attempting to demostrate.
It's a bit like the cognitive failure that led to "software engineering". Engineering was created to formalize the manipulation of the physical world. Applying it to information synthesis was always wrong, yet people did it anyway, and it led to spectacular failure. We now have agile development as a result. Maybe the same thing needs to happen to psych "sciences".
