I actually know of several places where a Town has a right of way on a privately maintained road. In some cases, the road was formerly a town or county way that was discontinued but a "public easement" remains. In other cases there is no public easement as such, but the town owns land that is accessed by means of the private road, and therefore has the same rights of passage as other landowners.
Unfortunately, I am not aware of ANY of these situations where a town has agreed to contribute anything whatsoever to maintenance. The closest case I can think of is one where a town owned a large tract of public land at the end of a road that had at one time been maintained by the town. Part of the road had been discontinued, but the town had not been maintaining an even longer section.
The other abutters decided to form a road association and tried to get the town to be a member. It was revealed that the town had been shirking their duty on the unmaintained portion for years, and the last I heard, I believe the Town was thinking maybe they'd better step in and resume maintenance of the whole road. (I have not heard whether or not they actually did.)
But that's the exception rather than the rule. Generally in these cases the town will refuse to contribute anything. I believe they claim some sort of public privilege or immunity.
You might want to try contacting attorney Mary Denison in Winthrop, as at one time I heard she was looking into the possibility of a lawsuit against a town that owned public land on a private road. I would think there might be a Constitutional argument regarding forcing private individuals to maintain a road for the public's use at private expense without due process or just compensation.
I do know that there was a lawsuit in 1970, Jordan v the Town of Canton, in which the Maine Supreme Court ruled that a law which created public roads with no public maintenance was unconstitutional because the public's use in the absence of public maintenance would inevitably destroy property access. Access is a property right attached to the land, and when that access is damaged, that constitutes a "taking," requiring both due process and just compensation.
So far, the Courts have not ruled on why this does not also apply to "public easements" on discontinued roads (or in your case, public rights of access to public property over private roads.) It has been assumed that since due process and just compensation are provided when a road is discontinued, that is sufficient. But that ignores the fact that every time a landowner repairs the access to his property and public use damages it again, there is a new taking with no due process or just compensation.