Ultimately, what these royal judicial reforms are all about is trying to create direct connections between the individual subject and the monarch, cutting through the various layers of subinfeudation.

So going to the Great Houses to ask for their help is kind of counter-productive, because it at least implicitly establishes the precedent that the Great Houses have a say in what royal policies will be in their realm. Instead, you want to assert that the Great Houses have an obligation as Lords Paramount holding their titles from the King, to uphold the King’s Justice.
As for the younger sons, you’d probably want to encourage some formal training, whether that’s by being sent to the Citadel or having them be tutored by a maester. Depends on what’s more practical.