How do you remove the Automator gear icon in the menu bar?

If you are starting a long-running process with "Run Shell Script", redirecting its output should be enough. For example:

long-running-process &>/dev/null &

The gear icon appears then goes away shortly after, I think it is better than nothing...

If you are using a shell script, you can add "killall ScriptMonitor" at the top of it.

You can install Bartender to manage and hide the menu bar items. You can either choose to hide it completely, or stuff it under a separate standalone menu bar.

The spinning gear originated from a core system app called "ScriptMonitor" which you can find in the following file path:

/System/Library/CoreServices/ScriptMonitor.app

If you own Bartender, you can manually add the add by clicking on the plus icon found in bottom right part of the preferences. The gear icon will appear briefly, but it should remain hidden while the processes are running.