In Iowa, there are gaps between some of the designated trails/paths that constitute the ADT. Some of the trails are crushed limestone, but many are paved. With that said, I consider our state to be above average for dedicated multi use paths that are paved.
I wouldn’t think the trail would be an easy task to accomplish on anything less than 35mm tires. Perhaps a gravel bike/rigid frame MTB with slicks would be the ideal bike, given the range of surfaces and conditions.
I have ridden many of the paved trails that make up the ADT in Iowa. Most are new enough to be in very good/excellent condition. I wouldn’t expect this the whole way, or even most of the way. I would figure most of the trail across the country would be crushed limestone, at least when you arent on roads that they needed to use to connect the trails together.
ETA- I know that part of Colorado is for sure not bike friendly.