I have to say, I was interested in your analysis, it inspired me to read the article for once. It seems your conclusion is quite the opposite of the one in the article. You're free to draw your own conclusions, but often when you see this there's something being left out.
As and aside, it's fascinating how effective the $50 is. It definitely does look like you get diminished returns quickly, though.
(From TA) "staying in housing they rented or owned"
I suspect there's something not being captured here. I'm wondering if there's somehow an option in between "staying in housing they rented or owned", and homelessness.
Perhaps, they're managing to staying with friends and family for part of it. I could see there being some gray area here. The morality of intentionally staying homeless to save money could be debated in various scenarios.
It's also ignoring a number of other dimensions of wellbeing that did improve only in the higher paid groups. Feeling safe where they slept, welcome in society, and maybe most importantly employment.
It's only if you look really close at one metric, which has moral judgements attached to it, and maybe explainable by living arrangement other than pure homelessness (keep in mind the criteria "pays rent or owns house") that things look so pessimistic.
Seems like the discourse on this topic is not permitted to involve the happiness, security, and likely long term outlook (due to being more employed, not just happy feelings). According to your analysis, we basically disregard the wellness of the homeless, and are only permitted to consider how it impacts us.
This is not an unfair critique, you mention some of these things, but they're handwaved away. I think you got the mental health aspect wrong, but quite frankly the results are confusing and a little mixed. The article itself doesn't do it justice.
It should be noted though that this is also trying to completely turn around someone's life, according to one metric for $6000 in 6-months. They know it's gonna end in 6-months. You can only do so much with this. I know you're using the term in the study, but this says nothing about universal basic income, or long term base assistances.
Also, I can't help but note all participants mental health was shown as decreasing, which is just odd. I double checked and it looks like this is 6 months from enrolment, so this is right after getting booted from the program. This is not a shock. People with mental health problems have issues leaving programs THEY paid for.