There are a number of problems. While it doe solve the "I-wish-I-Had-a-five-string" -problem, it would make learning the instrument harder, as;
*Teachers aren't used to it
*Printed fingerings in educational (and other) material won't work
*you would very soon have to find notes far away up the neck, something that is not easy for beginners or even intermediate players.
*Much orchestral repertoire would occasionally have some notes that would be very high in thumb position. Most solo repertoire would be nigh-unplayable.
The G-string is arguably the most used string on double bass - in classical anyways. For jazz, your main problem would be that you won't be heard, acoustiacally anyways - and those low strings are pretty hard worked. I'm clueless about rockabilly and other slap-styles.