As the error says, you have problems with the network dataset itself not the Route layer or solving the model. I'd try several things:
1) Add a network dataset to ArcMap and when prompted do not add any source features. Does it draw correctly and has edges? If yes, then the network dataset has been built properly.
Try running a simple test of routing between two points in ArcMap with the network dataset before taking the network dataset into a model.
2) Create a new file geodatabase with a new feature dataset. Use the Feature Class to Feature Class GP tool to convert the Streets feature class. Build a new network dataset there just to test it would work in a "fresh" environment.
3) Did you upgrade a network dataset from 9.3 to 10 recently? Do you store your network dataset in a geodatabase of 9.3 version or 10?
4) If the network has no edges perhaps it needs to be "built". Did you build the network dataset before using it (can be done in Catalog window of ArcMap or ArcCatalog)
5) I've seen this issue once with the proper network dataset - the problem gone away after installing SP3 for 10.0. Do you have SP3? If nothing helps, could be a good time to update maybe to the latest SP (SP5).
6) Also keep in mind that if the network dataset has none of the edges available then the stops you load have basically no place (no link) where they could snap to. Therefore you will need to re-load them and re-calculate their locations once the edges are in place.
Since you have lots of inaccessible parts of the network in your network dataset, consider using the query builder (Route Analysis layer properties window > Network Locations tab) to exlude locating on certain edges.