Only Two

Intersex babies are babies that are born with reproductive organs formed somewhere on the spectrum between male and female genders. It is normal practice in the US for a doctor to guess the gender at birth and perform gender assignment surgery. "For more than 50 years, the medical community in the United States has often defaulted to treating intersex children by conducting irreversible and unnecessary surgeries—and no clinic has firmly instituted a moratorium on such operations. Even after two decades of controversy and debate, there remains no research showing that early, medically unnecessary surgery is helpful to the intersex child." -hrw.org

That does break the either-or of the whole thing.

Why not leave them alone until they can be involved in the elective surgery on their own body?

"In July 2017, three former US Surgeons-General, wrote that they believed “there is insufficient evidence that growing up with atypical genitalia leads to psychosocial distress,” and “while there is little evidence that cosmetic infant genitoplasty is necessary to reduce psychological damage, evidence does show that the surgery itself can cause severe and irreversible physical harm and emotional distress.” They said: “These surgeries violate an individual’s right to personal autonomy over their own future.”"
" A growing number of doctors are opposed to doing unnecessary early surgery under such conditions. Practitioners also increasingly recognize the suffering of intersex patients who underwent the operations without their consent."
-www.hrw.org

Physiological changes like deepening voice, facial structure, muscle mass, and breast tissue.

I'm sensing a bit of limiting options from you. You want surgery to assign the chromosome indicated sex to the child before puberty. How capable are children up to age 11 to make decisions on surgery with their body with respect to their sex? Identify genetic sex and perform the matching surgery.

Looks like you're quoting the rate for ambiguous genitalia. Intersexequality.com estimates that as an inclusive group, intersex occurrence may be as high as two percent.
isna.org, soc.ucsb.edu, a story on Guevedoces from livescience.com, a pdf from interactadvocates.org all do not mention the effects of puberty.

hrw.org mentions "Undescended testes in boys, which is a minor atypicality of the sex organs, requires a simple procedure in early childhood to prevent future infertility."
A story on pbs.org talks about changes during puberty and the use of puberty blocking medication. The effects that are mentioned include changing structure of the face increasing muscle mass, deepening voice, and breast tissue development. Face structure and voice deepening may be the worst parts. Looking at puberty changes in general more inform this more.

The cultural and social issue at hand is how we treat people who do not fit into one of the two narrow types of sexes.

Hahaha

And you are classifying the two as what? Perhaps:
1. Gene basis: XX and XY (some malformations exist XXY and not XY)
2. Physical basis: properly formed reproductive organs (one in 100 do not fit standard male and female categories)
3. Reproductive basis: A male or female which can produce offspring with a fertile partner.

Anyway. There are two. Despite all the differences in physical formations and dysfunctions of the reproductive and hormone systems.
There are two despite the range of imagery, sensations, situations, or concepts that people are aroused by causing physiological response that is arousal.
There are two, and if you do not fit into my definition then you deserve to be treated differently than other people despite the fact that you are a person.

There are two.

Oh, it looks like everyone here has correctly interpreted your use of the term gender to mean sex and not the behavior and social aspects of the individual.

Gender is not a social construct.

Didn't he actually do that one?