What a great topic. I've read a lot about how illogical it was in GF3 for Michael to have given up the children; but until now I don't think I've come across a discussion about how and why that would have come about.

Here's my theory: At the beginning of GF3, when everyone is at Michael's home after the church ceremony, Michael says hello to Kay and tells her, "It's been a long time, hasn't it? Eight years." So, in the film's time period, he's referring to 1971. In 1971, Anthony was 20 years old and Mary was 18 years old. Since Mary is now legally an adult (I assume), Michael and Kay would have had no further cause to see each other, since collaboration on the children's welfare (visitation, schooling, etc.) is now over.

So this sounds like, after GF2, Michael gradually became more generous with Kay. If he didn't give her shared custody, he at least re-admitted her into the children's lives. Judging by the way he interacts with her in GF3, I'd say that at some point after GF2, he welcomed any excuse to see her, and what better reason than the children?

Maybe 1971 was also the year that the children began to distance themselves from Michael. Since they were now adults, they left Michael's house to go on their own or to live with Kay. This was probably the last time Michael saw Anthony, who, as we see in GF3, has bitter memories of childhood. Michael probably saw Mary off and on, since she seemed to be closer to him. But I believe they both wanted to form a closer relationship with their mother because they didn't see as much of her during their childhood. Michael was likely ignored for a while as his children's focus was on rebuilding the maternal bond.

When Michael writes in his current (1979) letter, "I entrusted your education to your mother for your own best interests," I believe he is speaking of their higher education and is referring to 1971 (or thereabouts), when both Mary and Anthony were of college age. But I also believe he is being disingenuous, as though he gallantly "gave" the children to Kay. In 1971, the children were adults; there was no custody to "give." Also, as adults, the children's education was now a matter of their own choice. In 1971, Michael's children simply left him, of their own free will, and Michael does not want to look that reality in the eye.

On a side note: I do believe Michael enlisted Kay's help when it came to Anthony's education; and it also served as another excuse (reason) to see both Kay and his estranged son again. In GF3, we see how Michael had his heart set on Anthony becoming a lawyer. But I can also imagine Anthony in 1969 as an 18-year-old, or in 1971 as a 20-year-old, rebelling against his father by refusing to even enrol in college.

As a result, I can see Michael, eight years later and in desperation, turning to Kay to get Anthony on the road to higher learning. Kay was an educated woman; she would encourage and convince Anthony that college was the right path. In Michael's current letter, education is used as an excuse to imply that it was haggles over college that separated him from Anthony in the first place, when in fact it was Michael and his mob life that did it.