If BB and/or Jolla can implement fine grained[*] permissions that overrides what the apps asks for (and they always asks for far more than they really need) then that would be a huge selling point for people who value privacy and security but still want to run the occasional android app but without the android baggage (ie google play services which uses up half the available memory, and your device manufacturer's own "services" that takes up the other half of your memory).

[*] the standard android permissions are too loosely defined, eg apparently to get access to the camera's flash LED (so you can use your device as a torch) you need to give access to the complete camera infrastructure such that the malicious "torch" app and also take pictures and videos. Another example, some media players need the "access phone info" permission which they can use to determine when there's an incoming call and so mute or turn down volume of playing media. But the "access phone info" permission also provides (I believe), the phone number, the caller id, etc, ie stuff that malicious apps shouldn't be able to access if all they need to know is whether there is an incoming call. Plus when installing new apps, google has now dumbed down the permissions info dialog such that it doesn't show all the permissions an app requests.