A Collet Chuck is particularly useful when an object is transferred between lathes. For example, a part can be started on a "traditional" lathe, held in a collet, and then moved to the rose engine lathe which can also hold the object in a collet. The use of collets to hold objects minimizes the error (runout) with moving to a new lathe.
For example, you can shape an object such as a shaving brush on your "traditional" lathe to the point where you can hold it in a collet. Finish the object there, including all the finish sanding.
Now the ornamental turning can be done on the rose engine lathe by moving the object to that lathe. If you are using an I-X collet, then keep the object on the I-X collet.
One ornamental turner with whom I spoke is working towards putting collet chucks on all his lathes (he has many). Then, he will be adding a stub to his traditional woodworking chuck, replacing the typical threaded mount. The traditional chuck would then be held by the collet chuck in whichever lathe he was using. This would minimize the runout when moving the object from lathe to lathe, whilst also not taking it from the chuck. (Note, for this, he is looking to use the bigger ER-50 chuck rather than the smaller ER-32 chuck.)
Some ornamental turners dispute the value of these. Their argument is that the runout experienced in moving objects between lathes is in the drive threads (onto which the collet is screwed). They argue that the more accurate means for transferring an object is to hold it in the Morse taper.