I'd be interested to know how this person considers it the next step up from oil pulling. As you say, their benefits are exactly what you get from oil pulling (which has been used for hundreds if not thousands of years). Teeth whitening depends on the oil, but coconut works great.
Of course the obvious answer is if you're interested in trying it - try it! Aside from the cost, I doubt there's anything harmful - and swishing with clay DOES have benefits. I know I do it myself if I have some kind of mouth pain or tooth infection. And for that it seems to work slightly faster than OP with oregano oil. So if when there's significant pain I use both and it knocks it out quickly.
The one caveat is I'm suspicious of swishing with clay on a regular basis. It is drawing power is amazing, but it takes the good and bad alike, making you need to supplement back in the good stuff. I don't know what specific nutrients/minerals the mouth needs, but I'll bet these people don't either. Clay is VERY depleting. OP is gentle and non-depleting and has been used for hundreds if not thousands of years with great results.
The second thing I'm suspicious of is that also because of clays powerful drawing action, any supplements you add to it stay binded in the clay and never make it into your body. That's why clay is such a good detox, it grabs the toxins, binds them and won't let them go until they're excreted. This is also why you should never take supplements any time near when you're coming clay - they just get absorbed by the clay.
So most of the additives they're talking about will never make it into your body.
Will it ever be BETTER for the gums than oil pulling? I couldn't imagine how. And after your teen years, cavities aren't as important and gums become all important for oral health.
Here are my thoughts on the ingredients:
Natural magnetic healing earth - clay
high vibrational energies - clay
coral calcium - why?
ground lemurian crystals - huh? probably found in clay
quartz - how does that help?
oxygen - could help if released from the clay (doubtful) but OP also oxygenates
ocean trace minerals - nothing specific to oral health and would never be released form the clay anyway
ancestral probiotics from volcanic microbial soil - clay