To be clear, this is just my opinion, and it’s based on my experiences with psychiatrists as someone with pretty severe PTSD and other mental health issues.
Neither of those approaches tackles underlying emotional issues. They’re popular because they give an easy-to-follow framework for “treating” very complex emotional issues. CBT is especially bad IMO, because it’s more like teaching people how to mask and distract themselves from the effects of emotional issues rather than actually resolving them.
EMDR is like a “one neat trick” approach to tackling trauma, as if moving eyes around is some sort of magic spell to reverse trauma. In my experience, trauma is treated through reprocessing traumatic memories, and while EMDR does focus on reprocessing memories, it’s a brute force approach, rather than a careful and measured approach, where emotional issues are slowly and steadily unpicked at a pace which is comfortable to the client.
EMDR absolutely can be used by professionals, but where I live, it is used exclusively and very rushed. You are given a pre-allocated block of, say, six sessions, where you are expected to pretty much immediately get into the traumatic memories from the first session, and if you are resistant, you’re told that you are wasting time.
That’s why I hate EMDR, because it is held up as the “evidence-backed” method and thus professionals are functionally forced into rushing it, because the insurance provider won’t pay for the likely months or years of therapy it takes to do therapy in a safe and healthy way.
Again, just from my own personal experience and that of my friends who have been through it.