Bluetooth is a convenient wireless technology for streaming audio from a mobile device and for communications, but it isn’t ideal for gaming. Latency can be rather long, which means you will always hear your audio slightly later than you see the visuals on screen.
Also, when using chat functionality Bluetooth limits the frequency range available, so audio loses some fidelity. Having wired audio for gaming ensures zero latency and maximum sound quality.
Bottom line: wired game audio and Bluetooth chat is an ideal setup for a variety of platforms.