I disagree with the reasoning, and I normally eat lunch at my desk because it's convenient and I get to watch crap on the internet.

First off, being the person that gets to put out fires is in no way a certain path to a promotion. And the experience gained from it is no different than if you just shadowed the person who would usually be fixing those problems.

Also, you mention how doing this you get an extra hour a day to do other stuff, does that mean you leave an hour earlier than your coworkers? That's definitely not going to get you promoted in most places.

I say take the lunch hour, at desk or elsewhere and spend it however you want, then work smarter for the rest of the day.