That is actually a good question - i will have to pay more attention to that the next time Part 3 is on (and i just checked and found out its on AMC later this week).

If i had to answer now, i would say its more a case of there aren't many left and Vincent seems willing and able to do the work necessary to run the family. Michael seems resigned to many fates in Part 3, and one of them is picking a successor - does he view his health issues as a sign that running the family is literally killing him? I don't think he had Vincent in mind as his first choice, but he felt more a need to pick someone then to wait for the right person.

But the next time i see the movie i might have a different answer.