There's a bunch of questions you might be asking here, and I've explored a few:
Why don't some species require more than two individuals to mate together before producing offspring?
Because putting three gametes in the same place is a lot harder than putting two, and isn't worth the extra bit of genetic diversity.
Why don't some species have more than two sex-chromosome makeups?
It's kind of complicated to make chromosomes work with more than two; when two individuals mate, how will the chromosomes be exchanged to actually create a viable karyotype every time? It could conceivably work under processes other than meiosis, though.
Why do species not express gamete stages?
Actually, there's an awful lot of plants that do weird things with gamete stages - look up "alternation of generations."
Also, this answer is helpful: https://www.quora.com/Are-there-any-species-with-more-than-two-sexes.
Apparently some species of lobster commensalist has some very complex way of reproducing, involving a lot of different genetic roles beyond just gamete and zygote etc.