All these postulated “facilitator” genders are implausible, because they get no benefit from their role in reproduction. The genders that produce the gametes are the ones that actually reproduce, the others are essentially sterile. The only way they could evolve is through some sort of eusociality, like with sterile worker ants and bees. You could consider the sterile female workers a third gender, since they are genetically distinct from the fertile females and fertile males. Workers are X0, females are XX, males are XY. Workers are produced when eggs germinate without fertilization, meaning that the workers are much more closely related to the queen than her sexually created daughters and sons. That means that it can make more sense for the workers to help the queen raise sons and daughters than it does for them to try to raise their own.
But keep in mind that the distinct division between genders that we see in mammals isn’t neccesarily the most common way to do things. Many, if not most, animals are capable of producing both eggs and sperm. Some produce eggs at one point in their life cycle, and sperm at another. But what is the difference between sperm and eggs? The main difference is that sperm are motile, and eggs aren’t. There are some species that are isogamous, meaning that they only produce one kind of gamete, a large motile one. But this is somewhat inefficient, since it is essentially creating motile eggs. Probably the first sexual animals were isogamous, but quickly evolved two gamete forms.
But why require two gametes to fuse to form a new organism? Why not three, or four, or five? Well, the real question is why you would require two, instead of one. Many organisms get by just fine without sexual reproduction. And it is something of a mystery why an organism would go to all the trouble of sex to produce an organism that is only half related to them, instead of 100%. The most common explanation is that sexual reproduction speeds up evolution and produces more variation. But there is a price, you children are only half yours. Sexual reproduction is worthwhile, but not overwhelmingly so. Requiring a third gamete is riskier and more complicated, and your children are only 1/3 yours. It seems that once you’ve got the simplest version of sexuality you’ve done enough to get all the variation you’ll ever need and more. Increasing that is only going to cause problems, and won’t solve anything that having two forms of gametes already solves.