I think Michael had no choice but to accept Vincent. His son wants no part in his crime organization and he sees no reason to drag his own son into this world even though he seemed disappointed that his son wanted to become an opera singer instead of an lawyer. Then Vincent comes along with the Corleone blood in him. Even though Vincent was the illegitimate son of Sonny, he needed someone to pass on the family dynasty and Vincent was the perfect alternative to his son. Even though he was another "tough guy" Vincent seemed like the only choice for him. Like his father, he has a certain sense of loyalty to the family as well and I think Michael really admired that.

The scene where they take the family photograph, I think that was the point where the story hints that Michael needs to accept Vincent as part of the family wheather he personally likes it or not.

There are many, many scenes in GFIII that Coppola wanted to serve as a reminder of the first two films: Not only the dialogue and the little tidbits, but adding in the flashbacks scenes as well, which really annoyed me the first time watching GFIII.