Briefly:
You're coming at the issue from a philosophical point of view, asking bigger questions about the "right" thing to do, so on and so forth. I come at the issue from a practical point of view - what can we do to ease people's suffering and make them safer, what actions has evidence shown to be helpful/effective when it comes to mental illness, how can we decrease stigma and enable mentally ill people to live a life they're happy with, etc. I'm an absurdist and believe that either life has no greater meaning, or that the idea of humans somehow being able to conceptualise such meaning is as ridiculous as trying to get cats to understand quantum physics. I believe that humans get a single life, and that the only way to make that life meaningful is to enjoy the shit out of it - and that everyone deserves a chance to enjoy their life and feel safe. It's what attracts me to clinical psych, because it's a job that's about helping people enjoy their lives and do things that they find meaningful. I don't think our two points of view really mesh well with each other. That said, talk about whether the lives of the mentally ill are less valuable because they are a burden on society creeps me out, because it's the logic that people used for centuries to imprison mentally ill people in asylums, forcibly sterilize them, shun them from society, and so on. I don't know what the answer is with regards to what the "correct" way to view the issue is, but imho you need to view the issue of mental illness with equal parts of logic and compassion. Looking at it from a purely logical standpoint is somewhat psychopathic, and looking from it from a purely emotional standpoint is ineffective (getting overly emotional while trying to help people with emotional dysregulation is like trying to rescue someone from a well while you're in the well with them). A happy medium between the two allows one to empathize and look at the issue from a humanistic point of view, while still being practical about what works vs. what doesn't and how best to help the person recover their quality of life.
Speaking of quality of life, it makes me uncomfortable that you're claiming that we don't know what to do with mentally ill people. While it's true that we don't have intricate knowledge of what causes and maintains the course of mental illness, we do know a lot about what helps it. Psychiatric drugs are one possibility, but psychotherapy is an extremely powerful tool as well. The most popular psychotherapies in use have performed much better than placebo when tested in double blind randomized studies, so we know that these therapies help people, even if we don't entirely understand why. The goal with mental illness is to reach a remission of sorts - therapists help the mentally ill to recover their quality of life and feel better about themselves, and psychotherapy can also diminish or completely eradicate distressing physical symptoms (eg. flashbacks, panic attacks, self harm, etc) that contribute to the distress felt by mentally ill people. It's incredibly important not to dismiss the life-changing effect that these things can have on a mentally ill person. Using myself as an example, I have gone from being house-bound and completely unstable, anhedonic, attempting suicide numerous times and having terrifying dissociative episodes - to having a job, doing well at university, having healthy relationships, and generally living a relatively normal life in which I experience moments of joy as well as sadness and anxiety. Drugs have never helped me much, so I have accomplished all of that just as a result of therapy. When drugs help people and they get therapy, the effect is twice as positive. I have seen numerous people make similar turn-arounds after being hospitalised for mental illness. So the lives of mentally ill people aren't...doomed, or pointless, and just because we can't "cure" them, it doesn't mean that we can't get them to a point that's almost as good as being cured. It would be better to think of mental illness as being similar to AIDS or cancer - we can't cure the people who have those conditions either, but we can give them treatment that makes the disease so weak that it's undetectable on lab tests, and allow them to live a much longer and happier life than they would otherwise. I don't think it's fair to discount a treatment that causes an improvement in quality of life simply because it isn't a cure.