Nintendo Switch has gotten support for Bluetooth headset

The latest Nintendo Switch update added Bluetooth support, only 4 years into the Nintendo Switch life

If you use Bluetooth headset, you'll only be able to connect up to two controllers wirelessly. If you need more, you need to disconnect the Bluetooth headset

You can pair up to 10 Bluetooth headsets, but only one can be used at a time. There's no support for microphone, this is only for audio

To pair a Bluetooth headset, go into Setting and scroll down until you see Bluetooth Audio (almost at the bottom). There you can click Pair Device and follow the steps