Check out this DMC thesis!

[Vergil Crosse]

Where was that ever stated?

The reasons why Arkham and Sanctus sucked with it was because of their intentions.

It was alluded to a couple times. Twice in DMC3 - Mission 19 where Vergil interrupts the battle between Dante and Arkham saying "I've come to retrieve my power. You can't handle it." and again when he states to Arkham "You of all people must realize you cannot control the power of Sparda." (Something to that extent)

Secondly, in DMC4 - Mission 20 where Nero tells Sanctus "I do know that Sparda had a heart that could love a human." (something akin to that)

So the conclusion that the full power of Sparda could only be controlled by ones who carry his blood line is not really unsubstantiated.