I don’t know which of the following plugins caused the problem:
3. /Library/Audio/Plug-Ins/HAL/InstantOn.driver (com.rogueamoeba.instanton-agent)
"I could resolve this issue finally with removing additional audio plugins in the folder /Library/Audio/Plug-Ins."
I just deleted everything in there, restarted and hey presto. Airpods and Jaybirds finally working perfectly and holding connection with Big Sur.
Press Shift-Option and click the Bluetooth icon in the menu bar.
Click “Reset the Bluetooth module.”
Press "Ok" to continue.
Code Block /Library/Audio/Plug-Ins/HAL
So now my Airpod's Pro is working property well.
Just remove this folder to trash and its done.