There is no "review board". If the code doesn't compile using only free software, it is rejected. F-droid is about pure open source.

Way too many random apks from the internet are loaded with 3rd party tracking, at the least. 

Anyone can setup their own f-droid repo and serve up their own apks.  Izzydroid, divested, guardian project, calyx, cromite, etc have all done this. 

Obtanium is basically a fancy curl interface. It doesn't verify anything and so long as the binary blob downloads, you can install it. Its no different than a user browsing and downloading binaries off the internet. 

F-droid needs lots of work, and it's a constant work in progress, but I feel it's vastly better than the alternatives right now.  

 Accrescent.app will be a much better alternative, protecting users from unsafe apps without making life hell for developers. 

 One can only hope.