OK, I've looked up the source (http://www.amazon.com/Testing-The-War-Weapons-Machine/dp/087...); this was a problem with early ones. "A friend of mine burned his hands to the bone pulling out the barrel of an M60 with his bare hands when the North Vietnamese infantry attempted to overrun his position in South Vietnam. He did not feel it at the time, but after the battle was over, he discovered the condition of his hands." They did add a handle later, and it sounds like by the time you tried one (and probably when I would have, if eyesight hadn't kept me out of the military).
Other points: Early ones had fixed front sights on the barrel.... Then he gets into reliability, and notes machine guns fire a lot more rounds than pistols and rifles. Some crazy guy put 10,000 rounds through a MG3 in 7.5 minutes, and 500,000 (!) through a FN MAG in one sitting. Obviously the soft sear is an issue there. He also says "a number of parts" can be put in backwards. Then he gets into ergonomics, the weight of the tripod (more than the less complicated one for the MG3, which also isn't made out of lightweight metal), the original bipod being barrel mounted (!!!) ... yep, your summation of it being a dog sounds spot on.
In all of my time in service, the only time I ever burnt my hand on a hot barrel was during OSUT, when a dickhead Drill Corporal had us all stand at attention after we'd formed up after being cleared from the firing line. It taught me a valuable lesson about how hot a weapon could get.
I have also seen an M60 barrel warp & bend after firing 1500 rounds non-stop.
As for putting the gun together incorrectly, I can see doing so in the heat of battle, or the fog (20+ hours a day on your feet, except while in the "front-lean-&-rest") of Ranger School, but have never seen such occur in normal training.
All in all, the M60 is still a venerable gun, and when proper tactics are employed (inter-locking fire) it is a more than capable weapon.
So much that, in one of my assignments, our platoon had 3 gun crews instead of the usual 2, because our PL had served multiple tours in Vietnam, and he knew the effectiveness of these guns - especially with 3 crews while doing a bounding overwatch.
"Fire a burst of six to nine."
The point is I should be able to pick up any weapon, take it apart and put it back together without knowing much about it. I, of course, was well trained. There's no telling how much training the next person has and we should expect that they will have little to none. I also spent time in a platoon with three gun teams.
Unless compared to its competition. In many ways its infamy is that while everyone else got it right (the Germans in 1942!) or at least a lot better, we didn't.... And I'd like to see what you say about the even longer list of problems in Wikipedia: http://en.wikipedia.org/wiki/M60_machine_gun#Design_flaws ; I not that it doesn't mention the barrel handle and bipod fixes, or the assembly problem.
The worst part was filling out range cards. (lol)
(I certainly appreciated the government supplied ~3,000 rounds/year of .22LR I shot on my JROTC rifle team.)
These barrels get so hot that, with night vision and at night, you can actually catch a glimpse of rounds traveling out the barrel. In retrospect is seems crazy to say that you can 'see' a round. Perhaps the night vision goggles are producing artifacts (not sure now).