To me, this process is common sense. How could I construct my argument to convince my company of its benefits?
You would need to look at how your company currently recruits and do your best to otherwise avoid breaking it. The more disruption you cause, the less receptive they will be.
The 1st challenge that many recruiters Google the candidate. I consistently get a LinkedIn view from a recruiter soon after I apply to most jobs as well as a hit on my personal website. You would basically need to ask the recruiters to give up their immediate Google search after they shortlist a candidate.
The 2nd issue is that many candidates are referred. At least at the companies my friends in HR have worked for, they went through the list together and pulled out the referred candidates for more than a resume skim. Obviously you need the name to process the referral.
The 3rd issue is that removing names requires more people to do more work. Someone now needs to spend time deleting the names from the resumes or at least covering them over. That also needs to be a different person than the ones doing the resume screen as otherwise, what is the point? The average resume is skimmed for about 7 seconds. This proposal would probably double the resume skimming time as it would take at least that to remove identifying information.
I am also not sure that this would change much. Sure, you can anonymize the names, but the university they attended can easily tell you a lot about their cultural background or race. Same with locations for jobs. About half the women I know have gender-specific achievements on their resumes. Men and women generally write resumes very differently, so even without the names you can often tell.
The place to start is to figure out how your company's hiring process works and identify the easiest way for anonymity to work within that process.