You’re not getting it. EMDR IS CBT, along with finger waving.
EMDR was started by Francine Shapiro as a treatment for anxiety after she was walking in the woods and noticed that after moving her eyes back and forth she felt better. She turned it into a therapy that involved having a person track her finger as she moved it back and forth in front of them, while also doing the standard cognitive behavioral therapy intervention.
If you do everything that one does with typical CBT and then add on the finger waving, it’s just CBT.
It should be a tip off that it turns out that Eye Movement Desensitization and Reprocessing doesn’t require “eye movement” after all. It evolved to be either eye movement or alternate hand/shoulder/leg tapping to apparently holding vibrators in each hand. Also, it turns out that it “desensitizes” for anxiety, but also for depression, bipolar disorder, conduct disorder…
When the critical element of the method isn’t actually critical, and when the method is developed for anxiety but is ultimately not specific to any particular problem, you have to start asking some questions about the method and mechanism.
It’s a bit like coming up with a treatment for treating a headache that involves giving an aspiring and stepping on the person’s toe, and calling it toe stepping headache treatment (TSHT). If there’s no difference in the effectiveness of getting rid of a headache between apsirin and TSHT, wouldn’t you propose that the toe-stepping part was not particularly needed?