The core experience as the Manufacturer intends. The same for TV's (I don't need SmartTV features but it's required for the core experience of certain models, for example).
The core experience for the Xbox One is Games, Media (movies, music, etc.), social networking (Skype, internet, facebook, etc.), voice and motion controlled UI and games, etc. Remove Kinect and you remove a core experience. Remove BRD and you only remove a distribution medium, all core functions still operate as intended.
The same goes for PS4. Remove the BRD and the only thing you have removed is a distribution avenue, no core functions of the machine have been compromised.
In other words, there is absolutely zero differences in functionality for an Xbox One or PS4 with or without a BRD. The only difference is how you obtain your games, music, video...
Remove Kinect from Xbox One and suddenly you have a vastly impacted experience, especially when you consider that once you make it optional, you have now impacted the experience even for those people that want it. As if it was option it wouldn't be used in even remotely as many games.
Regards,
SB