How should a PhD convert an academic CV into a consulting resume?

You shouldn't remove most things but you should change them. You need to reword them to emphasize and highlight more consulting-based traits. What you choose to put in and how you put it in the resume can make or break your application. If possible, you need to get additional consulting-related experience as well. That could be by getting leadership positions in various clubs, working on a side project, doing pro bono consulting (like through Impact Consulting) etc.