Just because the katra contains everything doesn't mean nothing is left in the body. Spock was able to function after transferring the katra which means his body still contained copies of all his memories. Chances are Spock's rebirth was a similar situation to Sim in Enterprise where as he grew he gained access to new memories but since a vital part was missing he was probably unable to integrate the memories until his katra was returned.
Besides, his katra was ALREADY downloaded back into HIS head at the time. If you recall, he only remembered that AFTER the process was completed, hence why he said it to Kirk afterwards.