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My experience is more with Department of Energy grants, so I don't know how much of this applies to NIH grants, but in grants I have dealt with a lot of time is spent basically promoting the expertise of the PI and how that person's expertise would allow them to successfully perform the research proposed in the grant proposal.
If you intend to strip out all identifying information then a large part of the introductory sections are can no longer be confirmed by the peer reviewers by means of checking the PI's background or references. Expertise and ability to successfully execute the proposed research is an important aspect of any grant, and it can be an unreasonable burden on a Project Manager to have to evaluate each proposal without some kind of expert input.
I agree with the sentiment of wanting to strip out all identifying information, as I have personally experienced bias from competing researchers during various peer review processes, but I believe that it would simply be impractical.
I've got a bad feeling about this.