Try and sync your BT headphones to your Xbox or PS4 or PS5, see what happens. You need a 3rd party dongle (which will introduce some form of audio lag, whether that bothers you or you notice etc is neither here nor there, it's happening) or Sony's proprietary headphones etc which don't use the machines BT, fuck knows how they manage it via the controllers but that's another issue.
I have some nice Sony XM3's here that I'd love to hook up, regardless of lag.
The companies have obviously used the lag issue as either a genuine reason so as to not piss people noticing lag off or an excuse to sell peripherals. Either way you ain't connecting your BT cans.
Nintendo have gotten a lot of heat for this, probably due to its handheld nature and more people wanting to connect their BT headphones. They've either said fuck it, enjoy your lag, minimal or no, or tweaked it in software and found a way to minimalise it to the point no one gives a shit.
Either way, it wasn't an option from the get go due to Bluetooth's limitations, not some Nintendo incompetency.