Yes, that is basically what you have to do. You might even want to use something like
rsync if possible so as to avoid copying files that already exist or files that haven't actually been updated.
Please note that if you do that, however, you'll want to make a special exception for the
<server root>/bin/shaderapiempty.dylib file. If you copy that file over again from the game, the server will end up crashing. So make sure you use the one from the srcds_osx download.
Alternatively, you could try using
SteamCMD to update the server instead. Insurgency is one the rare games that actually has all the Mac binaries needed in it's dedicated server app.