Now I see how to give the Innovation section a narrative, and accomplish other things too. The key was to see it as a way to reinforce the rest of the proposal.
The proposal begins with a Specific Aims page, which provides a summary of what I propose to do and why. This is followed by several pages of Significance, which provide a more detailed explanation of what the problems are that this work will help address. After the Innovation section comes the Approach, where I spell out each Specific Aim in detail, emphasizing how it will be accomplished.
In an old-style proposal, the Specific Aims page would be called the Summary, the Significance section would be Background, the Approach would be Methods, and the place now occupied by Innovation would be where the Specific Aims were listed, connecting the problems raised in the Background with the solutions described in the Methods.
There's no reason that the Innovation section can't also accomplish what a traditional Specific Aims section accomplished. So mine will begin with "I am proposing three Specific Aims, each innovative in both concept and strategy." Then I will have a paragraph for each Aim in turn, explaining how our approach differs from previous approaches and why it is the best solution to its problem. In doing this I will also be giving the reader an overview of the Aims both in the context of the broad Significance they've just read and in the context of the detailed Approach sections they're about to read.
An open letter to my fellow industry scientists: Why the March for Science must be led by us
1 hour ago in The Curious Wavefunction