As long as your PC/laptop has Bluetooth, you can connect directly via Bluetooth without an adapter. It worked for me without any problems (I have Windows 10 on my PC).

Not needed with bluetooth on win10. It is possible to connect. But the delay of the inputs is
the controller disconnects completely randomly.

According to experiences in the Linux community, it is not only compatible with the adapter (you need the drivers called XOW and XPADNEO under Linux), but according to my own experiences, the adapter is even absolutely necessary under Linux for many platforms. The background to this is probably that Microsoft is backing Bluetooth LE (which is basically quite welcome). If you use an older computer (e.g. like a 2017 NUC in my case) that does not yet support BT LE, pairing with Bluetooth does not work, so there is no way around the adapter.

Attention! For me, the controller does not work well with the PC (Windows 10 Bluetooth)! PC users should definitely use the adapter! (Or play with the USB-C cable)

The controller can be connected to the stick. It also works with Bluetooth directly.

Ich hoffe der neue Controller ist kompatibel mit Adapter...Bluetooth implementation unter Windows 10 ist der grösste trash

the question was probably a different one... Bluetooth also has a higher latency