Lightroom 6 has not been tested with Big Sur. Lightroom 6 has not been supported by Adobe for several years. It is out of date and completely unsupported. As operating systems change and advance there is no guarantee that old and unsupported software will continue to function. There is no version of Lightroom that you can "buy" anymore. If you wanted to upgrade Lightroom it would be necessary for you to subscribe to one of the creative cloud packages.
Note also that you can't reinstall LR 6 on Mac OS 11 (Big Sur), since its installer is 32-bit, and 10.15 and 11 don't run 32-bit programs. With 10.15, there were some reports that a previously installed LR 6 continued to run, and some where it stopped running. At this point, I think your only options are to reinstall 10.14 or subscribe to LR Classic (currently version 10).
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