^ IIRC, Delaware also signs the main route to/from Dover and the NJTP as following US-13 as opposed to DE-1 and I-95/I-295.
Delaware doesn't really sign any route as the main route. The DRBA has signage (advertising) to get to the CBBT and the Cape May Lewis Ferry, which instructs motorists to us Exit 4A to DE 1 (They apparently forget to omit that Exit 4A is on I-95 South!)
https://goo.gl/maps/YfxfbukWcsDXmYaR6The BGS for the 13/40 exit coming up is signed for Dover and Wilmington, as they are appropriate control cities for those routes.
But for motorists who are eventually getting to DE 1 and points south (across the C&D canal), no transportation department encourages any specific route.
And as you mention, if all traffic was truly following the freeway route, how much worse would traffic be on there?
US-13 is at grade, but it’s still a higher quality design and isn’t the worst. I’ve personally utilized both routes before, though usually just follow US-13.
In reality, US-13 should just be upgraded to a full freeway design between DE-1 and I-295, but that’s likely to never happen. It almost serves as a “gap” in the freeway system for that heavy traffic movement between DE-1 and the NJTP.
Much of this plays into the redundancy of the overall roadway system. The removal of any one route for any reason would cause traffic to be much heavier on alternate routes. Some people will avoid I-95 no matter what. Some people will avoid US 13 no matter what. Traffic, relatively speaking, evens itself out due to these factors.
As far as upgrading US 13 to full freeway, not sure if it was you or someone else that mentioned it before, but it would be nearly impossible on its current routing (for the reasons why someone like froggie uses it), and the 'gap' is resolved by simply using I-95.