It is supposed to be cut; however, not the way that it was. CDub, here's the instructions they were supposed to follow:
http://lib.store.yahoo.net/lib/yhst-24809172479195/cx5trailerhitch.pdf
(got it from this site:
http://www.mazdaparts.org/mazda-cx5-trailer-hitch.html)
Also, look at the price...$283 for the genuine mazda part. WITH wiring harness. So yes, you should have gotten it.
Also, the last page of the install instructions have a template for them to use for cutting the bumper; I would print it out, cut out the section that they were supposed to remove from the paper, and see how it lines up to your bumper.
The instructions also specifically say to "file it down to remove rough edges". I think yours was done with a hedge clipper...
Yes, you paid $350 for a bolt-on hitch installation that they butchered, and left out the wiring. I would be mad too.
Also, print this thread out. It can help your case if you show them that 20 Mazda owners are on your side

Good luck!