The statistic that says about 34,700 miles are in the state system tells you it includes the county route system.
Be careful with the mileage. Unlike most other states, West Virginia not only maintains almost all of the public roads statewide, but also many of the private roads. The most notable is the HARP (Home Access Road Program) with their unique version of a pentagon shield (house shaped). But there are a number of other categories of orphan roads and bridges that are state-maintained, including most recently MARP roads (Medical Access Road Program). These programs have not always been politically popular, but given that everybody quit maintaining private roads "after the one fellow who owned a tractor with a grader blade died in the 1940s" at least somebody is maintaining those roads.
HARP roads are public roads. WVDOH took easements as part of the process of accepting them into state maintenance.
The Medical Access Road Program was just a gimmick for using pandemic aid funds. They were existing roads already in the state system. In my county, "medical access road" funding went to paving a county route and repairing a slide on WV 7.
Anything that is WVDOH-maintained is a WV route, county route, HARP route, or Park & Forest route. DOH doesn't maintain other roads. While there are still a couple Delta route (state maintained with ambiguous ownership) signs around, those routes have either been fully taken into the state system or are no longer being maintained by the state.
On the flip side, there has historically been a lot of County Road mileage that was not being maintained by the State Road Commission (now DOH). Many of the state's backroads just simply lost population. I've got a relative that has a mile of Putnam CR-5 that runs on his cattle farm that hasn't been accessible to the public for more than 50 years. It finally got removed from the official road maps sometime in the last decade. The DOH is pretty good about cataloguing each route segment for budget purposes, so it is likely that the statewide mileage does not include the portions of the County Roads that are no longer being maintained.
If it's on a DOH county map, it's technically still a road and is counted in the mileage. DOH county map data isn't always accurate either - I've seen roads that the maps indicated were actively maintained gravel that had trees growing in the middle of the roadbed (Pendleton CR 28/6 comes to mind from about 15 years ago).
WV does have a significant amount of mileage in rural areas that is functionally abandoned and not passable by passenger car. There's also mileage that is a dead-end and just serves 1-2 properties that I've thought should be turned over to the adjacent landowners. I've advocated for DOH giving some summer college student interns a state car and having them drive the network, marking anything not readily passable for potential abandonment.