Shoulder seasons are really the best that's why the race is in April, and at that most years they miss some trail up north even before Flag. Only the slowest guys got to the North Rim late enough to ride the trail north of the Canyon. Most of the racers just take the highway up there.
Chunk? Like DH tech drops, there's some but not a lot. There's a lot of Chunky rocky stuff out there don't get me wrong, but there's a lot of HAB to go with it. Define unreasonable suffering, racers have posted pics of disgusting blisters, crashed and barrel rolled thru cactus, not to mention the whole lack of sleep thing most of them do. Riding at a more touring pace one should be able to have some control over the level of suffering, budget not withstanding of course.
Some parts of the trail get ridden a lot others not so much. The AZT 300 and now the AZTR for the 750 (there's more that you could add if you really wanted to) gets the most traffic out there bike wise.
There are a couple of segments that do get done a lot. Around Tucson SE side of town the whole passage is good XC mostly ST and sees lots of use from all user groups. On the E side off of the Catalina Highway starting on the AZT and finishing on Milagrosa
http://www.sdmb.org/trails/la-milagrosa/ DH chunk on the AZT as well as the social trail Milagrosa Canyon this is the DH run in the winter time.
East of PHx from the Picket Post TH south O&B for some major climbing on great ST this segment has tons of pics posted on line
http://www.aztrail.org/passages/pass_16.htmlThen you need to get above the Mogollon Rim to get to any really good stuff. Then the trail gets really fickle, snowed in in the winter most years and the dirt makes this sticky gumbo mud that stops you in your tracks. Early summer and fall would be the best time to ride for most of the stuff S of Flag.
Flag proper is in full season all summer long, they'll be getting snow soon and it just depends on the year if the highest stuff will clear in April for the race.
The stuff N of the Grand Canyon sees some traffic in summer, good riding but its a long ways from just about everywhere. There's a recent write up on the Kiabab
http://www.bikepacking.net/tripreports/az-sublime-kaibab/Hope that helps.
Tim