Who took care of the homeless person before?
The point is not that it will cure all societal ills, but that there are a substantial percentage of poor people who are not poor because they lack the ability to spend money wisely, but because they've been thrust into situations that are beyond their ability to solve without cash.
For starters, it is extremely expensive to be poor, to the extent where certain situations are far harder to claw yourself back out of than it is to remain out of them in the first place - but sometimes people get knocked down into that hole. Basic income provides a means to protect a lot of people from those kind of situations.
Basic income further changes the dynamics even for those who end up messing up: Aid towards those people can then focus on addressing the problems that prevent them from making proper use of the income, rather than be limited by their lack of income.
A lot of problems poor people struggle with are related: Mental health problems can trigger poverty, but poverty can also drive people far into depression and make other problems worse by virtue of making treatment difficult; physical health suffers, and reduced access to health care makes it worse; alcohol and drug problems are a frequent escape, and while they won't disappear with basic income, treating addictions without addressing the situations which contributed to driving people into addictions is vastly harder. You will not automatically solve the mental health and addiction problems, but you can reduce homelessness to predominantly a health/addiction problem (as it is in many countries with decent welfare systems, even without basic income) and leave yourself with a smaller challenge.
Consider that your argument is similar to that of the Namibian farmer in the article, who assumed that people would just spend it on alcohol. The reality is that while he was right about some people, he was also wrong about a lot of people, who were able to not only change their own situation, but contribute to further improve the situation of others (e.g. the money helped some start businesses, but also contributed to creating a market for those businesses).
It is irrelevant whether or not your hypothetical homeless person is still on the streets. The more relevant question is how many are helped off the streets or otherwise see their lives improve, and whether the improvements seen are worth the cost.