Removing "/Library/Audio/Plug-Ins/HAL/InstantOn.driver" (com.rogueamoeba.instanton-agent) fixed this for me. Followed this little trick worked for me too, now AirPods Pro connects properly to my Mac and they switch automatically between devices! Thanks a lot @Crazybob!

Same problem also here in Italy! Since I updated to big sur I noticed that my AirPods Pro continuously connect and disconnect from my 2013 MacBook Pro 15 and the don't play any sound. The remain connected for like ten seconds and then diconnect. AirPods Pro are working perfectly on my iPhone 8 Plus (iOS 14), iPad Pro (iPadOS 14) and Apple TV (tvOS 14) and also automatic switching works perfectly between these devices. Actually the Mac shows the Airpods' correct battery level, so I guess that is not a mere connection problem. There's should be another issue. Pls Apple fix this because I need to use AirPods Pro with Final Cut!