To give you an example, for snow plowing in Columbus: the state does the interstates, the city does SRs 104 and 315, even though they are state routes and freeways. The state has nothing like say Virginia, where everything is a state road. Here there are state, county, and even township roads. The township my mom lives in has a "road and cemetery" department and that's pretty much all the services she gets from them.
I think you might live in the same township I do. We find the zoning enforcement people are very much doing their jobs. (I once saw some township literature which summarized the budget: Road&Cemetery, Police, Zoning, and one other department (possibly Fire) all received roughly equal allocation. I didn't know the township even had a police force…)
Incorporated cities over some threshold (5K or 10K, I forget which) take over maintenance of non-Interstate state routes in Ohio. Signage of these routes is usually adequate; that may or may not be handled by ODOT in most cities, but there are a few cities which clearly do their own (cringeworthy but typically still functionally adequate) route signage.
I think US 23 was rerouted off High St in the 60s; I could pin it down tighter, but I'm not on my PC at the moment to look at the old maps. Columbus had seen explosive northward growth after the war, and whichever street carried US 23 would be burdened both with that intercity traffic and a lot of commuters. The one-way pair of Summit and 4th was better able to perform that double duty than High St, I suppose.
Moving the US routes to freeways in Columbus is something I can support, at least to some degree. I'll have to elaborate more on that in Fictional Highways.