Thank you for the reply, Neo_Institutionalist! The book was on a contemporary political issue. I'd rather not say more for reasons of anonymity, but I'm confident if I explained the book in a few sentences to you, you wouldn't be able to tell it was not written by a political scientist writing for a popular audience.
That makes sense about the academic press issue. One idea there is that by including a book chapter as a writing sample, it could show how the book was an example of rigorous research (though not primary research for the most part, as is the case with most books whether at academic or trade presses).
Thanks for the blunt feedback. It's made me think I should do more to emphasize the rigor of the book and my related research. While I don't know if it would qualify for, say, a top poli sci journal, I think I can certainly make the case that it's unusually rigorous for a PhD applicant.
If you have time, I am curious about your suggestion for independent research, as I'm currently considering whether to pitch a project to a political scientist at a top university, asking if I could do most of the work for the project but have their supervision (and if a paper comes out of it, co-authorship). And if that doesn't work out, I was hoping to provide volunteer RA work for that academic, doing whatever at all they need me to do. I didn't think academic scientists would take a fully independent research project very seriously, even if it were rigorously conducted.