No, this is not how everything works - ask Bernie Madoff.
A business makes a widget and convinces everyone that the widget has value, so people buy the widget. But the widgets don't buy other widgets, and whether the widget is bought is not dependent on whether other widgets are created. Here, in your scenario, other widgets/people with clout have to join and start buying to keep the other widgets afloat.
Now, assume the widget company is doing well. They do an IPO, and give the world stock options. People buy stock because they see the value in the widget. Their infusion of cash allows the company to make more widgets. But again, because widgets are not buying other widgets, the whole thing won't come crashing down if the company slows production -- they still have left over widgets to sell, and again, don't need widgets to buy other widgets.
Medium is different, because there is a subscription fee from the jump -- it's success obviously depends on more subscribers, but it doesn't need an endless stream of subscribers to stay afloat. It may earn less money the fewer subscribers it has, but would then just have to adjust costs. But in the thing you described, if I understand it correctly (and it is very possible I don't), members "purchase" other members, so you need a constant stream of members. That's classic Ponzi - having money flow up to initial users. The users below them then need users to "buy" them, so they encourage more users, and onward and so forth. But again, I may have misunderstood it all - completely possible.