There is no reason to think that private roads would not be managed in similar ways. Furthermore ownership is an important element to proper stewardship. Owners stand to gain from the proper maintenance, use, and development of their assets. If people owned elephants they would be taken care of and would not be endangered. Now of course, elephants are not owned because they do not have as much value but the reason why they are not owned does not detract from the point that if they were owned they would not be endangered.
Ownership also promotes innovation.
Telephone companies have developed phone cards systems for their public phones. Toll-roads have also developed ways of collecting tolls without even needing cars to stop. If roads were privately owned they would be developed and taken care of much better than they are under the status quo. Why would roads be any different. Surely road companies would be motivated by rational self-interest in a similar way so as to benefit them all.