Yes, Cheryl Cole was an effective judge for the X-Factor because she gave good criticism. If she weren't a good judge they wouldn't have invited her back. But fans are eager to have her back, and Simon Cowell also thinks that it's a good idea to bring her back. She is an asset.
I think that she was a very good judge on the x factor, and that they should have kept her on for a whole lot longer. She is one of the best singers in the world, and has a huge fan base that would bring in viewers to the show.
I do not believe Cheryl Cole was an effective judge for the X-Factor so I think it was fitting that she got fired. To me, the judges of the X-Factor are just as much a contestant and that commitment takes a lot of time and work. I don't know what the problem was but if I had to guess her heart just wasn't in the work.
Cheryl Cole was fired because the network's focus studies said that another direction would be better for the ratings. It was all business and nothing personal. If the X-Factor could have made more money keeping her, Cheryl Cole would still be a judge. If the new person doesn't work out, she may get another shot down the line.
The X-factor requires people who can recognize talent. She was only on the show because she emulated what could happen if a person was successful on a talent show. The rest of her career was sculpted by people who had a talent for the business. She had the talent to sing, but that's as far as it went.