I’ve got an old MacBook (13-inch, early 2015) that’s at latest version of Big Sur and Xcode,
and I just did a fresh install of CmdStan - no problem. see: 1 CmdStan Installation | CmdStan User’s Guide
I think you need to use XCode, and the xcode-select CLI, unless there’s some M1 Xcode issue - is there? I don’t know about Makevars and LLVM_LOC - (not an R fan) - but with XCode, llvm and clang++ are all under /Library/Developer/CommandLineTools/usr/bin
the problems you’re seeing sound similar to what happened a while back here: Upgrade to Catalina incurred problems building math lib dynamic archives (sundials) - #2 by mitzimorris