If your Mac remote doesn’t update (and it seems it hasn’t since May), simply download the installer from the Roon website and install it. This will at least bring you up to date, and then see if next time the built-in update works.
and install it over your existing copy. Your Mac remote has been on 2.67 since May while your server is on 2.71, and a remote that far behind the server can struggle to connect at all, so bringing it up to date is worth doing regardless of the update error.
One thing to watch for on a Mac: please drag the app into your Applications folder and replace the existing one, then launch it from Applications rather than from the mounted disk image. Running it from the disk image leaves the old version in place, which is the most common reason an update appears not to have worked.
Once it is running, please check under Settings, then About, that the Mac and your ROCK server are both showing 2.71 (build 1683). There was a hotfix released on 6 August, so it is worth confirming the server picked it up too.
If the built-in update check still fails after that, let us know and we will look at it separately.