Question Integrating wireless headphones with PC and stereo set - using external Bluetooth transmitter

So long after this his An universal wireless headphones / earplugs - a non existing audio hardware?, I have finally bough Bose QuietComfort 35 II Wireless Headphones. Unfortunately as it turns out, my Windows 10 does not recognize that my Gigabyte X570 Aorus Ultra rev. 1.2 motherboard has an integrated Bluetooth. Long story short: I do not intend to spend any more time trying to convince my OS, that I have a built in Bluetooth or to once again correspond with that laughable customer support of Gigabyte. Instead of that I will buy a new external Bluetooth transmitter of some sort, because my very old and almost-never-used one has no problem with showing up in that Windows and paring with these headphones - but produces audio of extremely low quality. [I am making an educated guess here: it is on the account of this archaic device working in standard 1.x or 2.x thus compressing data, because when connected through a Bose wire the audio in headphones sounds good]

And so until now my setup consisted of external Sound Blaster X-Fi Surround 5.1 Pro, plugged on one end to the motherboard and on the other to my stereo Panasonic SA-PM33 CD Stereo System. And I used to have in my audio and video player [i.e. Winamp and Media Player Classic] Sound Blaster chosen as an output device- but now I changed it to just default / primary system [thus to Sound Blaster as it is set in the OS as the main one]

But now The Question arises: should I use a Bluetooth USB-dongle connected to motherboard - or a Bluetooth adapter that uses mini-jack?

And in what standard and with what functions should that external device operate, so that I would not loose audio quality? For now I only know that it should have the frequency range of 20-20000 kHz

I would rather have a jack device. But is my stereo sending electricity through it to, to power the adapter? Or will the adapter also have to be plugged to a power source - or have a built in battery?

I went to some price engines trying to research the marker but as it turns out, jack Bluetooth adapters are a tiny minority and vendors rarely put info about the version in titles / links, thus figuring this out is not so easy

As it turns out, it all comes down codecs. And both the transmitter [like an USB dongle] and the device receiving the signal [like headphones] have to have the same codec in them. There are advanced [thus expensive] transmitters that show which codec they are using in a given moment- which is helpful as every model is able to use more than one and [depending on the receiver] the user could be in a situation of not knowing which codec is being used. And there are differences in both the audio quality and latency between various codecs

As for the dongle vs. jack dilemma. I have not planned this but I ended up using an USB transmitter, which sends the audio produced by my internal sound card. I also have a jack cable plugged into the I/O of motherboard, giving me the audio from the same source. And on top of that I still have the old setup of external sound card being connected to motherboard via USB and a stereo system being connected to that external device. All of this allows me for an immediate change of devices, with just one click- a special icon [made with freeware NirCmd from NirSoft] residing on a Taskbar gives me the ability to switch between sound sources thus eliminating the need to plug or unplug a transmitter. And in addition I do not have to worry about the battery life nor have additional power or data cable for the transmitter

As for other functions, some transmitters are seen by the operating system as an independent audio device. This might be important if the user is planning to use microphones build in the headphones and recording the audio from them with a software that have issues with selection of audio sources

Also problems can arise during the pairing process - some devices work flawlessly while other do not. This is unfortunately something that can be known on a trail and error basis and will vary from settings to settings, which is both time and money consuming

And the last important distinction of transmitters is the fact, than some of them are controlled by a smartphone app- which for me personally is ridiculous idea, because it forces me to use third [in my case] unrelated device to use two other devices [that I do not connect to my phone at all]