I thought it was because in DMC3, Arkham threw away his own humanity and therefore, his ability to love, and that's what made it so incomplete. Vergil couldn't awaken it because he needed the other half of the amulet, but even then, he too probably couldn't open it, because he had no love in his heart to protect anyone and only wanted power--a power he wouldn't even get.
in DMC4, the sword was already released with the amulets and Dante's bloodline put into it. However that's all that's been released; just its cosmetic form, but not the true power that comes with it. This can only come with the love for humanity and the will to protect not out of fame like Sanctus did, but to protect out of actual love from the heart.
This is one of the reasons I think Trish can use it and utilize its power.