I think it's already been covered but just in case.
Bluetooth is now a somewhat acceptable standard for listening to audio (with a bit of lag).
In 2-way mode with a mic, it's still absolutely godawful. Mono, and pretty much 'telephone call quality'. You aren't going to want to listen to game audio on it while chatting.
PS4 (and I assume Xbox) don't use BT audio profiles, they use 3.5mm into the controller, then send/receive as data from controller to console. Wireless headsets specific to the consoles use propriety wireless dongles and not BT at all.
But if you are Nintendo and are creating a console that can be docked or undocked and have god knows how many weird controller combinations including single joycon, I can understand if they've just segmented chat audio out to a phone app. KISS principle in action, rather than add 3.5mm jacks to every controller part.