Which class would run through these PQs fastest

. . .but the rank and kills you get on Nett's is irrelevant to the reward, is it not?

That is correct. Regardless of who you finish it on (or how quickly) you will always get the same reward room with the 50 boxes. I'm just stating the easier characters to do this on from my own experiences. I dont have 1 of every job (dont plan to), but from my experiences and from seeing my friends try it, these are the ones that'll make it easiest to breeze through.

If you are in it solely for the rewards and not for exp:
Use any character. It'll take you 15(?) minutes regardless of how fast you kill.

The character's I've mentioned almost never risk losing though unless you foolishly spam attacks on the yeti of course. And they also make the PQ much easier.