Can you ask for and legally fly routes, either direct or not, based on GPS information with nothing more than a handheld, or non-certified GPS receiver for the enroute phase of flight? Of course you can, as long as you're in a radar environment with ATC radar coverage. .
Absolutely not true. But you can look at your $200 GPS, get the bearing to xyz, say 271, and then call ATC and ask for a radar vector direct xyz on, like 271 deg. (wink, wink). ATC knows what's going on but if they approve the vector you're good to go. But technically it's a vector, not GPS navigation.