There are several different Intersect Faces options in SketchUp. The one most folks seem to use is Intersect Faces>With Model. I find in most cases that Intersect Faces>With Selection is more useful and often results in less clean up. Here's a case demonstrating this.
Earlier in the week a fellow sent me a model for a project he is working on. He was trying to get mortises cut in the curved pieces to take the crossing sticks. Fortunately in his set up of the model, he had copies of the curved component so mortises only need to be cut in one to make all of them.
I made a short video that hopefully shows the steps clearly. The key thing is that I copied the faces of the sticks and used them to create the faces inside the mortise by using >With Selected.