I have driven many Nebraska roads ... and a few of those mentioned ... you will be following the river valley of the Elkhorn River and not too far from the Niobrara river. The roads are generally pretty decent.
Not all the "towns" on the map are places where you can get fuel, food, or groceries ... but each of the larger towns have the resources you need.
I might mention that near O'Neill NE there is an amazing site "Ashfall Fossil Beds State Historical park" ... it is about 15 miles off your route east off US 20 ... it is an amazing place if you are at all interested in fossils ... the parking lot can accommodate your pickup and trailer. You will be amazed the critters that roamed the plains of Nebraska in times past ...
2004 Winnebago Journey 39W - 2012 Buick Enclave
Present @home Home: Oshkosh, WI
We call our rig "Ernie the Journey"