I have not driven one of the new 5.3's but have read and talked to people who say they are noticeably better/peppier than the previous models. I would still expect it to lag the 5.7 tundra a little.
As I understand it the AFM problems have been quieted down greatly, but if your planning on keeping it to high mileage I think the tundra would be a safer bet than AFM. My 2011 with AFM I sold to my uncle is still running strong at 75,000 with no problems, but it uses a quart of oil in a 5000-7000 mile change which I attribute to AFM.
I had a 2004 5.3L and it was a dog with anything behind it. Like redchevy, I have heard that the newer 5.3L motors are significantly improved in this respect. The 5.3L with max trailering package is rated for something like 11k lbs trailer weight, so I would think that it could handle heavy loads with the right gearing and transmission. But I'm not sure if that package adds an oil cooler or transmission cooler or anything like that. All I know for sure is tow mirrors, lower gearing, and a class IV hitch IIRC.
I think the newer Chevy's are quite a bit better than the ones from 3-4 years ago, which seemed outdated compared to the 2000-2012 models, which were average in terms of features/technology.
I'd have a tough time picking between Chevy and Toyota myself. As much as I commute, a Tundra just isn't an option. If gas goes back to $4+ a gallon, I wouldn't want to be driving something getting 15-16 mpg back and forth to work every day. I'd lean towards a Chevy, but it has it's limitations too. The Ram hemi isn't a terrible option and is more affordable than either of those.
5.7L hemi with 3.21 gears can only tow 8k lbs but easily gets over 20 mpg on the highway
5.7L hemi with 3.55 gears can tow 9k and does a little worse on mpg
5.7L hemi with 3.92 gears can tow over 10k lbs and will still be better than the Tundra on mpg