One of the biggest issues is that not every race can be every class. If you pick an undead you can't be a paladin, druid, demon hunter, or shaman. If you pick a goblin you can't be a demon hunter, druid, monk, or paladin. If you pick a pandaren you can't be a warlock, demon hunter, druid, paladin. If you pick a night elf you can't be a warlock, paladin, or shaman. If you pick a Lightforged you can't be a demon hunter, monk, shaman, druid, warlock, or rogue. Until/unless Blizzard changes their race locked class system then this could never be implemented in any way. Then you have the whole weapon enchant on melee weapons that are inferior to the death knight runeforging that would then require you to re-enchant the weapon once you swap out of the death knight into any other melee class.
Also as many people have pointed out, most people have alts they have already leveled and played for years now. This kind of system would make alts worthless...utterly worthless. Why pick up leveling an alt later on down the road if you can simply change classes on one character and have everything you need.