I live on the other side of Kansas and I haven't been out west recently but in general most small towns will have a gas station and that's it. If you're a middle-class looking white guy then you can probably get permission to pitch a tent in a church lawn or public park. (I'm not saying this because I think it's right I'm saying it because it's the reality of those towns.)
A lot of those towns you listed aren't even really towns any more, they're unincorporated and probably just a collection of houses and there will be no services. Tribune-Leoti-Scott City-Dighton-Ness City-Rush Center are all the county seat and basically the only towns in those counties and should have a gas station and maybe a restaurant and they're all about 25-30 miles apart until you get to Great Bend, which is a bigger town. Generally it gets more populated further east but then once you hit the Flint Hills after Newton you'll have a long stretch from there to Eureka, about 70 miles, where you won't be able to count on finding any services in those smaller towns.
The Newton Bike shop used to be famous for people crossing the state but it shut down a couple years ago and only does online now. I believe the Newton Fire Department took over their role of hosting touring cyclists but I may be mis-remembering.
Also, I apologize for the wind. Hope you get lucky and it's mostly southwest.