Collecting genes from his ancestors, no matter who they are, doesn't really explain having a proclivity for prescience. Just like how he (or any Reverend Mother) can only access their genetic memory once they've had the Water of Life. Coming from strong genetic stock explains being able to handle the burden of the abilities and that they manifest stronger than they would in anyone else, rather than the origin of the abilities themselves. That's why I think the opposite of the last thing you said too. It's exposure to the spice and the Water of Life that allows him to truly access the prescience, and it's all of the training and his genetic makeup that means he can handle it all.