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So yeah, I understand what the song is about, it's right there in the verses. But what's with the chorus? Why does "Well the cat's in the cradle and a silver spoon, little boy blue and the man on the moon" mean "If you neglect your kids they will learn to someday neglect you?"

It doesn't mean that. It's a reference to things that are typical around a baby, referencing that that was the last time he kinda knew his son.

"Well the cat's in the cradle and a silver spoon, little boy blue and the man on the moon"

The first part of the chorus you've quoted here all refer to items from childhood. They are, in order: a stupid string game, presents generally given as christening gifts in well off families (hence "born with a silver spoon"), two nursery rhymes.

Hmmmmm. Ok.

That's the ringtone on my phone for my Dad.