FIne-grained Volunteer Management
despite the detail provided it is hard to presume the workflow involved here but having looked through the Ride Scheduler tour I would approach building the core of this around Drupal Webform-civicrm and Activities and Views.

The webform approach gives the flexibility for capturing the info that civi core lacks.

The use of Activities or Webform delivers the notification process.

Views can be used to give displays of current 'requests' in a more user friendly way.

At this point I don't see CiviCase being necessary but that is because I am probably not imagining the workflow very well.

The one gap I see compared to RideScheduler is the automated call to the driver. But an SMS should be do-able without extra effort using eg CiviRules