The problem is not that your cats are bringing you presents. It getting up and leaving the house at 5:30 am in the morning!!! That is early. See if you were lazy like me and slept in - your stepping on presents problem would disappear.
Actually in my case, my cats use to have a cat door in the basement so they could come and go as they pleased. Then they starting bringing home presents. Which was tolerable - until they started bringing home live mice, birds and rabbits and dropping them at my feet! After I awhile I got tired of jumping up on chairs and closed out their little door. Now they have a automatic human door opener that opens and closes doors on command... Wonder who rules this household at times...