Of the people I met on the streets, only the very long-term homeless who did not have major mental health problems preferred to stay homeless.
For those, they appreciated the simplicity of life and had acclimatised to it and also understood that to get out of it is several mountains of effort (1 to get off the street, 1 to get to the same standard as the average person in poverty, and finally another to have a good life in a lower-income bracket). Those people were able to consider all of this and actually would choose to have a good life with nothing at all and just to be at peace with their situation.
Those people are rare.
The short-term homeless (< 6 months - seasonal variation means that there are vastly more of them in the summer) I felt made up a greater proportion of those currently homeless certainly would not ever choose to stay on the streets. They had real difficulty surviving. They're also a lot less visible, the short-term homeless are very fearful and will hide away from everyone and not interact with the long-term homeless and would mostly refuse services to help them as they lived in a state of denial. These are the people most likely to end up in the most horrific circumstances, they are the most vulnerable.
The other significant tribe is the one I disclaimed in the first paragraph... those with mental health problems. You cannot underestimate the effect of being alone all the time, living in some degree of fear and being hungry and unkempt. It rots your sanity slowly, it leaves you unable to do what seem externally to be simple things. You could not give these people homes, they couldn't cope with them, but likewise they're not coping with being homeless either and their hygiene reflects this.
So to the original question, are the homeless made up of mostly people who have no desire to stop being homeless? No, not at all. It's certainly true of some, but others simply do not have the ability to make those choices, nor the courage to accept help or the mental wit to comprehend and then deal with it. Whilst there are some that will never leave the streets through choice, there are far more who will never leave the streets through lack of choice, and then a large mass who will leave and for whom being homeless is the darkest part of their life.
It's just really important to not judge the whole by those who have the confidence to engage with others, that they are doing so probably means that they are already an exception and not the norm. I clearly don't reflect the vast majority as I never let go of the smallest sliver of hope, but most have and cannot imagine tomorrow being different from today. They really need hope given to them, and the support to help them help themselves.
As much as this does dominate part of my past, I actually don't think it's one of the biggest social issues. I once met an ex-drug dealer on the street who had been in jail and raped in there. He told me of the great respect he had for real victims (of which he didn't include himself), and that he respected those able to deal with childhood sexual abuse far more than anyone else he ever encountered and found their ability to respect themselves to be overwhelming.
Me, I find the homelessness in my past a source of motivation and some shame, yet the people I have the greatest respect for aren't those who are in that situation still or break free from it, but those a notch higher. I have the greatest empathy for those not living on the streets but who instead are confined to live in poverty. Poverty is a far more widespread problem that deeply affects far more of society. Homelessness is such a small extreme, and there are outs... but there are few outs to poverty for the vast majority. If homelessness is the frying pan (small), then poverty is the bush/forest fire (enormous) and it's effects on the people goes very deep and gets passed from generation to generation.