Good trouble shooting procedure.. You have narrowed it down to the coax from the switch to the roof **OR** the connection at the switch (It is possible some idiot hooked the leads up wrong, hey, I know for a fact it happens, though in the case of this RV it was a different set of leads hooked up wrong).
If you have to replace the coax there may be an easier way to route it.. But I'd have to have eyes on to be sure.
But if at one end of the coax you have 12 volts and at the other end zero, that cable is likely bad.. however
Disconnect cable from switch/power supply inside... SHORT it
Then go topside and see if it measures shorted if not SHORT that end, and go back to the switch and check the other cables.
If you need to run a new cable... You might try routing "overland" on the roof to a spot just above the switch, and then dropping the line straight into the wall.. If needed use one of those very long drill bits they make just for that job. Put good quality compression fittings on both ends and a good bulkhead connector at the roof. caulk it in well, weather boot the coax and caulk that well to make absolutly sure it does not leak.
You may wish to use a right angle adapter at the roof to prevent a sharp bend in the cable,, And you may wish to upgrade to RG-6 while at it (Won't make much difference but for the cost,, Might as well, every little bit adds up)