From personal experience, subjectively speaking, large doses of MDMA are psychedelic in very much the same way as the classic psychedelics like mushrooms and LSD.
It's a subtle effect, dominated by the stimulant and empathogenic effects, but it's decidedly there, especially when you close your eyes.
I suspect such doses are well beyond what's used in therapy though.
From what I learned (no personal experiences) the MDMA wouldn't be suitable for any kind of therapy? It makes for few hours of energetic ecstasy followed by few hours of lethargic depression. Without psychedelic effect on patient what is there to work with?
You're probably speaking of an extasy pill (which is often mixed with other stuff etc), not crystallic MDMA. Set and setting does a lot as well. MDMA can be a very powerful psychedelic (personal experience).
Since this article talks about MAPS, here is a MAPS source on explaining it [1].
So it's a mild psychedelic at best and not a psychedelic at worst (excuse my English, non-native, not an intended value judgement).
And the truth probably is that the word psychedelic is used for attention grabbing purposes only. I get that, but I do want to make that clear.
I'm simply summarizing the source.
[1] https://maps.org/news/multimedia-library/5545-mdma-the-movie...