First off, I again commend you for this effort. There isn't a lot of activity like this on Medium.
As for the slow response, there are a few things that may be happening:
1) It takes awhile to read a book. I'm about 85% done with my next book for this feature (according to my Kindle). But I had to move that book up in my queue in order to get a review done for this feature/pub. That isn't something I will normally do, but since I have two books here, it seemed only fair to push out another review ASAP, which I will do as soon as I'm finished. I will also add more in the future, but they'll end up further back in my reading queue and will just happen when they happen.
I used to read about one book per week. As I've gotten older, I just don't do that anymore. I don't even come close. My vision isn't what it used to be, and even good books knock me out like they're hard core sleeping agents. There's just no way for me to change this physiological truth. I also read other stuff: The New Yorker, Medium, other mags. I finish a book whenever I finish it, but it's never quickly these days.
2) Medium is still not a fiction-centric place. I realize there are non-fiction books in the list, but mine are fiction and so my expectations are low for any kind of Medium help from a promotion standpoint. But it all helps some, which is why I think participation is useful, and you never know when something like this feature might just take off out of the blue.
3) The reviews are done exclusively by other writers who are on the list. Maybe expand that to Medium readers in general? How? I have no idea. If I was good at marketing Medium stories or pubs, I'd make a lot more money at it than I do.
4) There's nothing wrong with frequent newsletters by you that have little more to say than a reminder to review a book (separate and not published on Medium itself). We all signed on for that, and you have a right to remind people to honor their commitment. It doesn't have to be any more than, "Have you reviewed a book recently? If so, thanks. If not, please do."
5) More articles might help, but not if they make the publication/feature too busy. I wasn't aware that after we post a review on Amazon that we are also expected to post a review for this pub until you told me in an email. I'm sure it's spelled out plainly that this is so, but I obviously missed it somehow. You could add a reminder of that requirement to your "pester" emails, which I think are an important part of this.
My last suggestion is to get on a Discord and Slack channel for Medium. The Medium Discord channel run by Robin Wilding is a good one. Not a lot of fiction writers there, but a few. The Illumination Slack channel has a lot of noise and clutter, but it has a lot of people on it. Again, probably not a lot of fiction people. But everything helps.
If it was me, because I'm so challenged for time, I'd probably just limit my efforts to very short weekly pester emails to everyone who has signed up. If people are annoyed by that, they shouldn't have signed up. And maybe the occasional longer email when you have time to report on the feature's progress.
And after that, just patience. Most people probably read books a lot faster than I do these days. I know I did at one time, but they probably also have stuff queued up in a list.
Just my two cents. Not sure if this helps or annoys! :-)
