16mm and Super16 are different only in the frame size. They don't even look different (frame dimensions aside). And the Hurt Locker was certainly shot on 16mm.
In terms of Red vs 16mm - both are completely different cameras. It's like asking DSLR vs Shoulder-mount. The answer is: it depends. What look does your film need? What budget do you have? What time constraints do you have?
16mm is faster in that there's practically no electronics. On a film camera you essentially have on and off and that's it (maybe sometimes a video split for the Director). On Red there's a million different settings and options, plus a 90-second boot up time, issues with crashing etc. As well, 16mm camera batteries last longer (again as there are much less electronics in the camera).
Also, in terms of that clean, high budget look - you can certainly get that out of S16. What stock are you shooting on? How are you processing it? I've seen Fuji 500T stock rated at 400 which is unrecogniseable situated within the midst of 35mm. I've also seen Kodak 50D which looks like a bad student 16mm film (probably because it was..) It all depends on your level of skill. 16mm you need to know your stuff, you need to be confident you can nail takes as you may not have the budget for hundreds of rolls of film. You need to know how to use a light meter properly. With Red and digital, unlike film, what you see is what you get. Whilst you may be recording raw on a Red, you can still get exactly the look you created on set. With 16mm, you essentially use the viewfinder as a frame composing tool and everything else comes down to hwo you lit and metered and how it gets developed.
I think they're both amazing formats and I love shooting on film. I also love shooting high end digital (though Alexa much more than Red). It really comes down to a preference depending on the story. That's why you hire a DP so they can make the decision about what's the best format for the story.
To me, whilst I find that Red can look great, it's still inherently digital looking. There's something 'magic' about the grain of film that I just can't get enough of.
Anyway, it also comes down to the grade, so whether you go 16mm or Red, you would be wise to invest in an experienced colourist to grade your film.