I'd be happy to begin. First of all, I will say that I think “competent” should be clarified to make it clear that it is intended to mean trained and knowledgeable on the subject matter of the investigation.
As far as independence is concerned, I think the concern that I would have about independence is that, when I think what you're seeking is unbiased, I'd be worried that independence would suggest that, for some reason, organizations have to go external to the organization, because I don't think anybody internal is going to independent. Many investigations can be done very effectively by internal people who are trained and who are knowledgeable in the subject matter.