By checking files and documentation shipped along side (Realtek's) driver provided by ASUS, we can find firmware file of rtl8761bu inside the rtkbt-firmware folder.
The original information posted here was wrong, after research we can find ASUS BT-500 Bluetooth dongles are using a rtl8761bu controller rather than some Broadcom one. Information about the chipset inside this product is nowhere to be found on the Internet. Yet I find the firmware for rtl8761bu among the folders, so that chip is probably the one inside this BT-500 model. I'm going to buy one dongle with same controller to prove this. Feel free to correct me if I am wrong.
PS: Simple devices like Wi-Fi/Bluetooth dongles will usually work with free software as long as it's main chip (SoC) is supported by the driver. Occationally the driver will work for similar chips.
PSS: On Github/Gitlab and many other coding platform, you can find many third-party drivers for popular chips. Many are blobless, especially those without a firmware directory and doesn't contain files like *.cab. (*.bin files are too ambiguous to determined directly)
PPSS: The scan issue's cause has been found, it's an unpatched bug. See:https://askubuntu.com/questions/1370663/bluetooth-scan-doesnt-detect-any-device-on-ubuntu-21-10
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