I like to organize my music by genre. It would be nice to have a %g tag available in the Mass Rename tool to move bands/albums into a genre subfolder.

There is %g (or $genre), the same variables are available in everywhere they are used, even if they are not listed, and for most fields you can just use $fieldname.
Be warned that songs may have multiple genres, in this case %g will be replaced by "genre1, genre2, genre3".

In the attached image, you can see how I added %g to the folder pattern. In the preview window below, mass rename tool does not recognize %g or $genre as a field.

Am I using the tool correctly?

It seems the songs in your screenshot do not have a genre, so it's replaced by the empty string "".
Also if the path doesn't begin with a slash, it will be relative to the current path of the file, which means you may easily miss the genre lost in the middle of the long path.

Oh, you're right. I see I failed to click the plus sign to add a genre. Thank you, it works great!