Thanks for the responses, all -- best to confront the uncomfortable reality now rather than deeper into my career. I guess that I took it pretty hard realising that I didn't have a feasible path -- sorry if I came off as brash or condescending, or if I didn't communicate clearly.
My experience has been primarily with later career-stage grants, so that must have distorted my perception. When I worked with PIs and graduate students on their grant proposals, my work was typically focused on the research dimension (e.g., preparing figures and reviewing specific aims). The biographical sketches in the grants I read had also given me the wrong impression.
This is, I think, critical. I (wrongly!) framed my original question in a very dichotomous "either/or" manner, when I realise now that I was really looking for options further toward the research end of the continuum. In my case, GRFP is out, but I'd like to think Hertz and NSDEG are both still worth a shot.