Not sure about HDG Select, but having had a LNAV fail in Europe a few years ago (737 Classic) while en-route, we were offered Vectors, but instead flew VOR to VOR ,with VOR/LOC, worked a treat, as an aside, the VIP's were were taking, one had a radio and was listening to our ATC conversations, they were rather impressed with the FO that day, got a nice tip at the end of the trip.
Whether you can dispatch, well that's a different ball of wax, if you ferry old aircraft, there are times the FMC will not have the database you want in it, if its not GPS, it a Classic, then it can be interesting inputting the VOR/DME and NDB's into the FMC.