If your mental health issues have utterly ruined your life to the point where you are homeless, that fact can help make you suicidal. To the degree that mental health issues can lead to suicide, I think it is reasonable to suggest they can be deadly.
Alcohol also kills germs. I loathe alcohol. I had a nightcap pretty much daily for about a year while doctors acted like I was crazy and failed to figure out how to treat my condition. I began getting better. I stopped drinking alcohol when I found other treatments that worked better.
I'm convinced I would have died had I not consumed alcohol for a year.
I knew someone briefly with a mental health issue who lived in fear of winding up homeless. She also drank alcohol very heavily for a time and felt enormous guilt about it. But she had surgery for something at some point and when they opened her up, they discovered some badly infected organ. They removed it and marveled at the fact that she wasn't dead.
I think her heavy drinking saved her life. After that surgery, she was able to get clean and sober, in spite of her mental health diagnosis.
However, many people use alcohol to self-medicate for legitimate issues. The fewer services available to you, the more appealing alcohol and other drugs seem. Alcohol is, after all, cheap and readily available.
Sometimes it is as simple as trying to be able to withstand a cold, wet night. Sometimes folks are in pain (a variety of things can cause this, from tooth decay to sore feet to serious issues). If you cannot afford psychiatric medications, alcohol is a way out. It might not be ideal, but it is available. More available than health care in many cases. After all, you are speaking of folks that don't have actual shelter, a proper toilet, sanitary conditions, or a place to store or cook food regularly.
I'll add that simply being homeless can be a deadly condition as can a myriad of things that you might go to the doctor for. Seriously, if you simply couldn't get anything to treat your condition, what do you think your options would be? What if daily life made you absolutely miserable and you couldn't find a way out?
And that is the entire point. Not having access to the things "normal" society does makes things different.