It's worse than that imho.
I live somewhere that provides a fair bit of support and free services (e.g., transitional housing) for the homeless. That doesn't get rid of the homelessness, all it means is that the remaining people without access to shelter and showers are those that are generally incompatible with polite society for a variety of reasons. (Crime, severe drug abuse, severe mental illness, etc.)
Implementing it as "anyone can hop on wherever" implements a weak filter for the worst of the homeless population using it as shelter. Those who've been refused or kicked out of available shelter.
I could see it working more as a "free transit passes" program so they could be revoked and not re-issued for people that were going to cause ongoing issues in the same way someone that shows up to transitional housing and gets high and assaults other tenants and neighbours could be removed.
You can say all you want about Castro, but he did in fact eliminate homelessness in Cuba.
The city provides studio apartments with "daily meals, laundry, showers, housing application assistance, community and health service referrals, skills building, relationship building, connection to volunteer and employment opportunities, and other support services".
It's not a home, but it certainly means that the people availing themselves fully of the opportunities presented to them aren't unwelcome on transit because of their inability to maintain basic hygiene.
I agree that potential problems should always be anticipated. But if the conclusion is "well we have homelessness so we can't offer a service that will be taken advantage of by homeless people in unintended ways" we can just as well agree that everyone with a six figure income move into a gated community and leave the rest of society to THE PURGE.
The obvious conclusion for a self-respecting society should be "let's try free public transit and also throw money and mental health support at the homeless population until they don't exist any more."
The objection isn't that homeless people will use the service, it's that they'll effectively make it unusable by everyone else.
Yes, in an ideal world that wouldn't happen. But whether or not we should have homeless people, we do, so proposals need to take this reality into account. If you think you can "just fix" homelessness, then do that first and worry about transit after.
But there are a lot of wrinkles to this kind of thinking. What if there is access to shelter, but it imposes some kind of deal breaking restriction like no drug usage? Or no felons? Let's get even more controversial - what if the shelter bans sex offenders, should homeless sex offenders be forced to shower in libraries? Things get complicated, and a lot of politicians aren't willing to die on these hills when their housed constituents have strong emotional reactions to these things.
The only way to reduce homeless is rebuild traditional value and traditional family structure. Less people get divorced, less kids get abandoned. People always get last resort from the big family and the church and family friends and church friends, etc.
Get help from the government and charity organizations should be rare cases.
But we are on the destructive path for a long time. Part of it is natural consequence of social change, but it's also partly due to some people think they are capable of design a new society from top to bottom.
Afaik, most abandoned homeless kids are queers, most often trans. Then kids with behavioral issues (yes, stemming from crappy parents in the first place). I really dont understand how return to traditional values and family structure would improve either.
The higher prices of the paid compartment would go towards subsidizing the free compartment.
The problem is political. At least in the US, this approach would be opposed by both major political ideologies: the right because it is a government hand out, and the left because it is unfair to have a "poor" section of the bus (see the reaction to mixed income apartments having a separate section 8 entrance, or restricting the amenities that section 8 residents are allowed to use, which is essentially what you are proposing).
Further, the nature of bureaucracies in our economics system (be they for profit or government) is to prioritize profit centers at the expense of cost centers, so it would be difficult to maintain the "poor" compartment to an adequate level. Compounding on this is the fact that the poor and homeless tend to be the least politically powerful groups, and maintaining adequete funding for this scheme would be difficult,
If homeless people do not have ready access to shelter, but do have ready access to public transit, then you get them spending time on public transit for the shelter it provides, which means they use it a lot more then they would if they were using it for transit. Public transit isn't even particularly good shelter, but since most places don't allow loitering (with selective enforcement against the undesirables), you again get them congregating on the few places that do.