How to change the iPhone's audio output back to non-bluetooth? (internal speakers)
This is a bit of a guess based on an assumption I am making (dangerous, I know). 

My assumption is that when you're playing games such as Angry Birds Rio that the audio is not actually outputting via Bluetooth at all. 

Instead, I think the reason you do hear it when switching on a Bluetooth speaker is because at that point the iPhone starts outputting via Bluetooth. Prior to that, I think your iPhone is outputting via the iPhone's internal speakers but you're not hearing it because the volume level is not up for games. 

What I mean is that iPhones have multiple volume levels. For example, it is possible to have the ringer volume level all the way up while having the games audio level all the way down. 

As a test, I would start playing one of the games and make sure the volume is turned up (and also that the mute switch is not on). If that solves the problem, great! If not, report back so I or others can dig deeper.