What makes it confusing is when the discussion compares the same framing versus the same distance to subject.
If you need more reach, and the camera stays in the same place, adding a TC will increase the depth of field. Only if you move the camera backward so that you have the same image as without the TC, will the depth of field remain the same.
This is exactly right, on both counts, both the source of the confusion and what happens.
So, for wildlife where you merely need more focal length and can't get closer, you get more depth of field with the TC. For a portrait, if you already have a good composition, its very unlikely that you would add a TC and then move backwards, so that is not a common usage of a TC.
There is actually a very common usage of a(n effective) teleconverter that works exactly that way.
All those variable-aperture zooms out there? They're basically using a variable teleconverter. So, if you're doing portraiture with one, you might well take a waist-up shot wide open and then zoom in (with the lens, not your feet) for a head-and-shoulders shot still wide open, but (obviously) at a smaller aperture. And, even though it's a smaller aperture, it still works out to a smaller depth of field.
It works out to exactly the same smaller depth of field as if you had just taken a single waist-up picture and then cropped out the head-and-shoulders portion and enlarged it to the same size as the original waist-up shot.
If you're using a constant-aperture zoom, the depth of field is even shallower still, because now you not only have the enlargement factor decreasing depth of field, but you've also got the larger physical (or, at least, apparent) aperture at work to boot.
Somebody posted to some other thread an online tool that graphed background blur as a function of focal length, aperture, and distance from subject. Wish I could remember where it was, but the uptake is that a 50mm f/1.0 will have the most blur for things near the subject, and a 400 f/2.8 will have orders of magnitude more blur for things past about ten feet or so. So, if you want the least-recognizable background possible, speed wins for tight spaces but focal length blows it away if you've got the working distance.
It works the other way, too. If you want to maximize depth of field rather than minimize it, you want a short focal length lens stopped down as much as possible with as much distance as possible between you and your subject.
...and then you can use a lens with movements or adjustable aberrations to further complicate matters in either direction....