I find it interesting that they went the way of the county building the road and signing as an Interstate and county highway, where Phoenix's loop roads, which were funded primarily via sales taxes in Maricopa County, are all state routes. Is NDOT that dysfunctional that Clark County residents wouldn't trust them with the money to build out their freeway system? Or is there some political reason that prevents county funds from being applied by the state to the county in question?
Yes, it is interesting to sign the non-Interstate portion as C-215 instead of NV 215. And I suspect that NDOT is constantly balking at providing the necessary funds to complete the freeway loop, because Las Vegas.
It wasn't an NDOT dysfunctional issue. The initial idea of building the beltway came primarily from Clark County and/or the Regional Transportation Commission of Southern Nevada, not the state. Clark County's 1990's "Question 10" voter-approved transportation funding initiative included (among other things) a modest gas tax increase which was specifically dedicated to fund the Las Vegas Beltway construction--a similar Question 10 circa 2006 extended/increased this funding. This was funding that NDOT would not have been able to obtain as easily, especially when taking into account needs elsewhere in Las Vegas and in the rest of the state. Also, IIRC, they were able to save some steps/time in the planning stages by the county building it, since the funding source was primarily county and not state (or federal)--I think (but don't quote me) it had to do with less stringent/complex environmental impact studies since FHWA wasn't directly involved.
I could also speculate that a political reason for NDOT not building all of the beltway would have been that an increase in the state gas tax or some other measure would have been required in order for NDOT to take that on. Northern and rural Nevada would likely not have been keen on a tax increase that would likely not have benefited their transportation infrastructure whatsoever. At that time with the growth boom in Southern Nevada, NDOT was already spending more money/resources in the Las Vegas Valley than anywhere else statewide.
NDOT isn't necessarily balking at funding the completion of the beltway. It's just that their funds have been somewhat spread thin amongst other capacity project priorities on roadways that they currently control: Multiple I-15 widening projects in Las Vegas, US 95 widening in Las Vegas, I-580 extension, I-80 in Reno, US 50 & US 50 Alt in northwest Nevada, etc. And keep in mind that primarily NDOT, not Clark County, built the existing beltway interchanges at I-15 and I-515 in the south.
The difference is Phoenix is the State Capital and the Center of Population at the same time.
In Nevada, the capital is way up near Reno (Carson City) but most of the people live at the extreme south end of the state (Clark County). So there's a constant political battle between the northern part of the state (which is historically where everyone lived because, before A/C, Las Vegas was essentially inhabitable) and the southern part. Almost everything pits North versus South, and usually North wins those fights.
Actually, the reason for Nevada being mostly settled near the Reno/Carson City area before Las Vegas was explorers went through northern Nevada first (en route to the Gold Rush), and northern/western Nevada was where all the gold and silver mines were first found in the mid 1800s. People lived in Las Vegas well before air conditioning was mainstream.
The state is politically charged between the north and the south, but I wouldn't necessarily say that the north always wins those fights--there's a lot more legislatuive representation from the south, so some would argue that things tend to skew in favor of the south more often than not.