Then people will make a complex device that hooks in where the LED/LCD grid does and converts the signals intended to drive the panel into a video feed for recording. And making a device that directly takes the output of a Bluetooth device's DAC (or even grabbing the digital output directly) should be even trivial.
It wouldn't be easy, and would need some adjustments for colorspace etc but this is a method I have seen discussed online before.
The speaker part is much easier. At some point in almost all modern audio systems, there will be a PCM signal as part of the DAC step. Hook in here, pipe that signal to a computer's audio in / microphone port, and you've got a pretty clean audio stream.
