Personally I don't think Doc would sign off on coaching his own son.
I'm not sure if I would go that far, but he has deliberately kept his distance over the years. He's made it very clear that he's wanted to be Austin's Dad, not his Coach (kind of the antithesis of Todd Marinovich's father).
With that said... Austin will be what, 20 years old when he's drafted? Maybe Doc figures that he's been "just Dad" long enough, and that their relationship can handle that other dynamic.
If Rivers is available when the C's draft, they have to take him. He's not perfect. He's a little small to play the off-guard, and he might not be a natural 1 (although I'm not sure if his father was a "classic" PG, either; Danny Ainge certainly wasn't).
But, he's got the potential to be a game-changer. At the very least, he's a poor man's Monta Ellis, which would be pretty [dang] good to get from the #15 pick.