I will say there is a possibility, and I love this theory. Think about this for a second. do you know how when you walk up to the stuffies if fnaf 4 and C.C. says these are my friends? well what if he is refering to the souls possesing his stuffies and not the actual animatronics. we know the puppet is a prize counter worker, and stuffie versions of the animatronics can be won as prizes as seen is UCN, so what if C.C. saw the kids being murdered (hence the golden freddy stuffy saying you know what happens when they catch you,) and also saw the puppet put their souls into the stuffies, and got the stuffies. This also explains the happiest day minigame, where the souls of the kids are at a birthday party and them put on masks of each of C.C.'s stuffies. (a metaphor for their souls being put into them) this means that the i will put you back together is left unexplained, but, as you said, it could be refering to reincarnating him as gregory. (maybe he was always named gregory)
but heyyy, thats just a theory, a Game Theory!!!