Neither does the PS5 but there’s a 3.5mm jack and a built in mic in the controller. Sony’s claim is that standard Bluetooth isn’t fast enough to stay in sync with the screen and your control inputs, so they have their own bespoke wireless (but not Bluetooth) headset. I’m happy enough with a cheap pair of wired headphones from the supermarket.
Just chiming in to add that Xbox controllers also have 3.5mm jacks, although they’re annoyingly incompatible with Sony’s 3.5mm Jack to get the mic working (at least last time I tried with a pair of old iPhone earbuds which use the same configuration as Sony).