At a local, "let's give it a try" show you may find a variety of breeds but when points and money is on the line you will only see QH and, to a much lesser number, Paints. Those horses are specifically bred for the job.
That is not to say that an Arab or TWH hasn't done cutting and maybe they were even good at it, beating QH at the local level but they just never make it to the statewide level or beyond. As a breed, they just don't have the build or cow sense that has been bred into the QH.
The right tool for the job - you can pound in a nail with a pliers, but a hammer does a better job.