As far as I am aware the following is relevant,
The last update to Lr 6 was in Dec 2017, there will be no further updates.
The Lightroom 6 installer is a 32 bit application.
The following recent applications released for macOS Catalina, Big Sur and Monterey will not run 32bit applications, so the Lightroom 6 installer will no function. If the application was installed prior to your update it will likely run.
You may need to investigate if it's possible to revert to macOS Mohave.
Regards, Denis: iMac mid-2015, 5K 27”, GPU 2GB, Ram 24GB, HDD 3TB, macOS 11.7.4 BigSur,; LrC 12.2.1, Lr 6.2, Ps 24.1.1,; ACR 15.2,; Camera OM-D E-M1