LR6 won't work under Big Sur (or Catalina) I'm afraid.
I'm writing this on a 2008 MacPro running El Capitan and it is still fast enough for editing even video. However, of course I can't run the latest versions, but I do have a more modern MacPro ready to take over from it that will run Catalina at least when I find time to do the transfer.
On this machine I have a license for all of the Adobe apps and on my laptops I have the Photograher option. I was annoyed at first by the idea of a subscription, but at just under 10 pounds per month for both LR and PS it seems pretty good value. CaptureOne has an option for a perpetual license, but it is getting on for 300 pounds. I think if I wanted an alternative I'd go with PhotoLab from DxO especially given how much I like Nik filters: in fact I have a license for it as part of a package, but normally use LR/PS and Nik and occassionaly CaptureOne as it seems better suited to PhaseOne files.