Also, you've got something more than slightly wrong about the police being unable to find Neo. The stolen car would have been reported, likely found, and Neo was bleeding all over the place all the way back.
Add in the fact that he took the time to enjoy a joyride, which, in an open-topped car is basically asking someone to be able to ID you, and open-topped cars aren't exactly things people forget about within the course of a single night, especially not ones with such interesting occupants such as Guts and Dragonslayer, to say nothing of Neo.