What they don't cover is... Okay, say you rob a bank on Purge day... but can you really spend that money? I mean if you deposit in a bank that was robbed and they know it is their money, can't they just legally take it back? You would be immune from prosecution for the robbery, but surely they could take their possessions back.
Like... if you stole your neighbor's car, you couldn't be prosecuted for the theft... but the next day, he could just take his car back because he can prove it is his... Couldn't he?
So hurting people is the only crime you could get away with since it all happens on the day and the next day it's just done.
Although... I wonder... if you killed them on Purge day but just left the body in your backyard... couldn't police find that body and charge you with ancillary crimes related to possession of the dead body?
I don't think the concept was really too well thought out.