You might also get some comments from climbers in the Sierra Nevada who would call that chute on Browns "easy class 1", reserving class 3 for smooth vertical walls
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I have only downclimbed that chute on Browns, having ascended the peak twice by a brushy northern slope that also involves a couple of airy moves high on rock. I recall the chute being exposed, but only a couple of spots where the holds seemed marginal at best. Most of the time, there was enough room to stand or sit, like a series of ledges. Perhaps some 10-foot segments of that chute could be rated class 4. I tried ascending that route once with my wife, but found the looseness of the lower chute unsettling, so we bailed.
Having now a lot of experience with peaks all around Arizona, I think that the chute on Browns is still one of the hairier chutes I've ever dealt with. I had always been told to just take it in 10-foot segments and don't look outward. It has severe exposure.
As far as whether Class 3 in Arizona is tougher than elsewhere... that is hard to quantify. I think that once you get used to the type of rock in a certain region, the classifications may be technically correct, but the nature of the rock - its composition, tendency to flake, form nubbins or hangs, erosion patterns, and so on - can add a whole unquantifiable dimension to classification. While I feel comfortable on class-3 in AZ, I may not so much on class-3 in WA, since the rock up there just lays differently. The little nuances I have grown used to in AZ may not translate to other places immediately. So I'd feel uncomfortable at first.
Try approaching Browns from the north. Go about half-way up the slope toward the chute, then angle left and traverse through the trees to a rock fin, then up the slope behind it. There should be a rough path. At the very top, you will have to move along rock with easy moves, but lots of air all around you.
I commend you for doing this during this time of year! I have grown soft and stay indoors when the temps are above 110F (like now). I'd like to explore that part of the Mazatzals when it cools so keep me in mind.