Bluetooth headphones connect via Bluetooth adapter with Switch, PS4 and PC?

I recently bought new Bluetooth headphones from Sony (WH-1000XM3). I would also like to use this for gambling as well as for my computer without having to connect a cable (my PC does not have an internal receiver). I then informed on the internet and learned that the PS4 recognizes only certain headphones and the Nintendo Switch generally does not accept. Remedy would probably make a USB Bluetooth adapter, which is then plugged into the computer, PS4 or Nintendo switch dock and the headphones could thus be connected.

Now my question: Does each Bluetooth adapter the same purpose or are there differences and things that I should pay attention to before buying? For example, there are specially designed for the switch USB adapter from third-party, but then I do not know if they then work well on PS4 and PC. I also found one on Amazon for 10 euro.

(https://www.amazon.de/CSL-Bluetooth-verbesserte-Energieeffizienz-Technologie/dp/B01N0368AY/

Is this suitable, or does it necessarily have to be one that is 20 euro more expensive, but achieves the same effect? Or. Is the extra charge worthwhile to achieve better latency in sound transmission?

As far as I know that does not matter at the switch what you dransteckst although the internal bt chip it can still abkann 😑 in ps4 I have no idea.