Text in a carousel is a no-go; pausing when hovering is a neat idea, but not possible on touch screens, nor easily discoverable by the average user. Even if the timing is long enough, I still feel stressed out by the fact that I know the text could be dragged out of my sight any time, which stresses me out so much that I can't concentrate at all on the content.
The second terrible mistake is using page indicator dots (which do a good job of indicating the page) for navigation, let alone using them as the only means of navigation!
When you take into account these two things, a carousel can actually become comfortable to use and may no longer enrage its users. But that's still a far cry from being better than showing the content in a regular list:
> I've used them and my clients and the customers of those clients were happy with them.
What do you mean with "happy"? Did the customers actually understand how to use the carousel, did they discover its content, did they click on it? Not being annoying does not suffice to qualify as useful.
The second terrible mistake is using page indicator dots (which do a good job of indicating the page) for navigation, let alone using them as the only means of navigation!
When you take into account these two things, a carousel can actually become comfortable to use and may no longer enrage its users. But that's still a far cry from being better than showing the content in a regular list:
> I've used them and my clients and the customers of those clients were happy with them. What do you mean with "happy"? Did the customers actually understand how to use the carousel, did they discover its content, did they click on it? Not being annoying does not suffice to qualify as useful.