I can't speak for MDMA, but I have experimented (ok, recreated) with psilocin (shrooms), 4-ACO-DMT, and DMT, which are all serotonin agonists.
The best I can describe it is that a high enough dose feels like putting your brain into debugging mode and then cycling through hundreds of circuits simultaneously. It's not so much "I am experiencing happy now" as "I am experiencing happy, sad, confused, interested, pulsating colors, fractals, childlike wonder, fear, and every other cognitive state simultaneously or in rapid succession."
Some people get very spiritual about it, but I think you kind of have to be a spiritually inclined person anyway to have that specific experience. I always saw it more of giving the hardware a good jolt and seeing what falls out. It's pretty common to go from laughing uncontrollably, to inconsolable sadness, and back again, fairly quickly. This is where a lot of the fear about "bad trips" comes from, but I've found you can manage that with practice and having experienced people tripping with you or babysitting.
I do think there are therapeutic benefits though. Even if there are difficult periods, I find I always felt happier, more ok with the world and open to new experiences and attitudes for up to weeks after words. The article briefly mentions how there's a hypothesis that depression is possibly an effect of individuals who "over learn" negative experiences and become fixated on negative outcomes, and this makes a lot of sense to me. Aside from a sort of "purging" feeling just from such intense bursts of emotions, I could see how it helped to kind of kick me out of the traps of negative and habitual thinking and emotions. Maybe it helped my neurons "relearn" there are other experiences, emotions, and modes of thinking that I can access too?
This is all anecdotal evidence though, gathered while under severe impairment, so take it with that caveat. I would bet that aside from the legal issues, the hardest part of investigating things like psychedelics in a scientific way is that every experience is inherently subjective and personal. Trying to describe and talk about them gets very squishy very quickly.