The biggest thing with Bluetooth audio is the codec used between the headphones and the source. The most basic is SBC which nearly everything supports, however it's pretty low quality and high latency which makes it inappropriate for gaming.

Pretty much every other option involves licensing fees, and there's nothing really universal. Apple's stuff uses AAC and more recently Apple Lossless, Sony's stuff tends to use LDAC, anyone using Qualcomm chips gets AptX sometimes in HD and lossless variants. If you don't match on codecs, you end up on the shit-tier SBC codec, which isn't fit for purpose here.

After licensing, they would need to write support for all these codecs into the playstation sound system and to get the latency down to an acceptable level for gaming, you might even need hardware codec chips, I'm not sure.

Bluetooth audio is the wild west, and given all the above I can't really blame them for not supporting it

Edit: out of curiosity I had a look, there's LHDC, LC3 & Samsung's own one now on top of everything else