They have the perfect amount of speed with a friendly mind. They're ready to work without being jumpy or excited. They can work cattle unlike any other breed.They can do almost every event and make it look good.
All of the reasons already listed and I grew up with them. My grandfather started breeding QHs in the 50/60s with a son of Poco Dell, then also added in a son of Poco Tivio and a son of Poco Pine. My first mare of my own, I was 7, came straight off the King Ranch. I was on a QH before I was born and after I was born, I was riding before I could walk, literally. I've met & owned some that weren't the cakewalk stereotype but they have all been very cutting bred and true to that, they can be very sensitive and extremely responsive. With them you have to be a rider, never a passenger and always thinking 10 steps ahead of them.
I have to second the big butt opinion too. Love me a big, powerful hind end. My littlest cowpony at 9 months...I like big butts and I cannot lie...
And Woodstock when he is fit makes me drooly. Right now he looks like a broodmare, he came through winter a bit fluffy. He had a major tendon/soft tissue injury and finally I can say he's been sound for a significant amount of time and will be back into boot camp after I'm back from vacation.
I love Quarters because all of the ones I've worked with have been beyond forgiving and gentle in nature. We used to have quite a few but they have passed away due to old age (one being 50 years old! O_O) or sold them on for various reasons. We still have two though, and a hand full of half QH mares who are worth their weight in gold.
This guy was my favorite in terms of build and atheleticism. He had quite a bit of showing behind him and was a breeding stallion before we gelded him for the sake of his quality of life and because we just didn't have the time to put into a stallion. He made a fantastic gelding and an even better ranch/performance horse. He was sold on to a boy in college to use on the rodeo circuit and I've heard he's doing quite well.