I know this is quite old, but did you ever get a solution? I am having this problem now too where networks that are completely linked are filled in.
Let's get Jay Sandhu to weigh in on this again, but perhaps you could provide more details (like screen shots) of your problem. While I'm not as active in NA as I used to be, it's been my experience that service area polygons are more of a guideline than a hard, fast boundary. What really matters is how the routes themselves show up as the service area.
What software/version are you using and what settings are you using for the service area. As Joe mentioned, post a picture of a problem and that might help figure out change in settings.
Jay Sandhu
I started a new discussion here https://community.esri.com/thread/208963-service-area-polygon-appears-to-fill-in-between-the-roads-f...
Thanks!