So I'm attempting to build some fairly large routes used for cross country shipping. Like for example if we wanted to avoid an entire city. Most polygons are running afoul of the 2000 street intersection limit imposed by the routing service. Is there a way around this? Maybe a restriction type that'll eliminate some smaller streets from consideration to get me under the cap?
If you're using the routing services from logistics.arcgis.com (like through the Ready to Use services in ArcMap or ArcGIS Pro), there is not really a way around that limit (to my knowledge). You cannot create or modify any of the restrictions on those services, and there is no restriction on there that would weed out smaller roads.
Your best bet for using the services would be to solve for shortest time, which, when solving at the kind of scale you are (cross country), will force hierarchy. Hierarchy will do the weeding out of the smaller roads for you, as that is one of its purposes. About network analysis with hierarchy—Help | ArcGIS Desktop
You could try converting some of the polygons to line barriers, which would give you an additional 500 roads out of the equation, but on the scale of going cross country, I'm not sure how much 500 more roads restricted will help.
If you're going to be solving for routes often, you may want to consider creating your own network dataset or purchasing StreetMap Premium for ArcGIS | Street Dataset | Geocoding Addresses , as network analysis with your own network dataset does not have a restriction on the barriers.
Hope this information helps!