I know first hand[1] that visibility of an Android game is basically zero if it's not on the Play store. So something like this could help alternative ways of distribution to become feasible.
1) Distribution - I don't want to take responsibility for distributing random APKs that could have sleeping viruses in them (that I don't catch). And with something like F-Droid it would first link to a download for F-Droid, and then there'd have to be some sort of collaboration to deep link correctly into F-Droid from an install.
2) IP Infringement, Copyright, etc - Like you mention on your blog post, Google is coming down pretty hard on apps that infringe on copyright (I also know from first-hand experience) - and I don't want to be a party to that. Seems like a good way to get a ban.
Your game looks very nicely polished though. It really sucks that it's not allowed on the Play Store (like in so many other similar cases). IP companies appear to be making a fortune these days by just sending out automated take-down requests for things like this - no (real) appeal, just poof and months of your work disappears into the void!
What F-Droid needs (maybe it already has these things too?) is:
1) It's own URI schema for these things - so that you can link directly to F-Droid without a Google Play link. So https://f-droid.org/repository/browse/?fdid=in.p1x.tanks_of_... should take you to the app if F-Droid is installed, otherwise you should be prompted to download it from the web page.
2) A way to deep link into the app when it's not installed... so that if the user downloads and installs the app, the first page they see is the one they wanted to see (e.g. a particular app).
Unfortunately I haven't once seen Google go for Fair Use in the multiple instances I've been involved in already. The truth is, once you're big enough to have contacts in Google, then you can start to appeal. But if you're small fish, you're gonna fry. Don't want to take any chances on this one.