OK, well it's definatly in my opinion that 2 things are evident between GF2 and GF3. #1, Michael is definatly alone at the end of GF2, meaning no kids from the way it's taken. #2 he has shipped them away by GF3. IMO he starts his quest for redemption by the last shot of him a little older at the every end of GF2. He looks sad, alone, forelorn, it's a powerful imagine that's engrained in my head.

Michael's capacity to take over for Vito was one of Genetic and Environment ("Education"). At the beginning of GF1, from what he told Kay and his knowledge, he might as well have been in Sonny's shoes. So him assuming the reigns was a function of this.

Additional to this, Fredo's death olbviously was his remarkable sin. He slew Fredo, who is probibly the most innocent and sweetest character in the series for a weakness of character, that made him, who he was.

Therefore, it's my speculation, that he realized his mistake and allowed Kay to take away the kids, by doing this their "education" would be a decent one, not emersed in a Mob world. Not one like the one he showed at the wedding (knowing the doings of his Dad, bandleader, and knowing all about Luca).