Common ancestry doesn’t require that there was only one initial type of living organism. It just means for example that all mammals descended from a common ancestor. Or all tetrapods descended from a common 4 legged fish.
Once a bone has mineralized it should survive. Oldest mammal fossil found dated to 125 mya. Instead of rabbits in the Cambrian how about elephants or other very large mammals in the Jurassic? We know there were mammals in the Jurassic but they are all very small.