Gender neutrality is the concept of getting rid of gender roles and stereoypes. It has nothing to do with erasing gender itself--rather, it deals with making it so anyone, regardless of gender, is not hindered by gender roles or stereotypes.
For example, traditionally, dresses, makeup, and having XX chromosomes is equated with being a woman. Likewise, physical strength, sports, and having XY chromosomes is equated with being a man. Gender neutrality would make it so that anyone could wear dresses and makeup without being judged by others.
Basically, gender neutrality is the aim to end the whole "sugar and spice and everything nice" vs. "snips and snails and puppy dog tails" idea, because anyone should be allowed to be who they are and express themselves without being considered strange or abnormal.
There would be no difference between girl's toys or boy's toys--kids could play with whatever they wanted. Dolls wouldn't be considered a "girl" toy. Trucks wouldn't be considered a "boy" toy.
It's an idea that helps people of all genders, not just male or female. People who are nonbinary, or who don't identify as strictly male or female, would be included. Intersex people would be included. No one would be restricted into two genders, but beyond that, no one would be expected to act or dress a certain way just because they're a certain gender.
Gender neutrality aims to separate gender from sex. Gender, in reality, is something your brain creates due to the society around you and the rest of your identity. It shouldn't have anything to do with your biological sex. That's why it's silly to say things like "there are only two genders: male and female" because gender isn't biological. It's mental and cultural. It's not even correct if you say that there are only two sexes, because like gender, sex is a spectrum. It's possible to have both testicles and a vagina, or to have a penis and a uterus, or to have a vagina but no uterus, etc. And that's not weird or strange or abnormal. It's just a fact: bodies and genders are separate and very diverse.
What you're talking about at the end (shapeless clothing, boyish bod, or a no pitch voice even categorize you as neutral) isn't "gender neutrality", but androgyny (sort of). Androgyny is both a way of expressing yourself and a gender. It means to be both male and female, or to express yourself as both male and female. And there's nothing wrong with that. You're right that anyone of any gender can express themselves however, but it's not okay to call the concept stupid or silly, because people should get to express themselves however they want without criticism, as long as they're not hurting anyone.
It's not an attempt to make their lives any harder. People ask things like to have others use "they/them/their" pronouns because it's part of what makes them comfortable and helps to reaffirm their identities. It's how they want to be respected. The only reason identifying as androgynous would make a person's life more difficult is that the people around them refuse to accept or respect their identity.
Oh, and there are more genders besides just "male, female, and androgynous". Lots more. That's part of why gender neutrality is such a good concept. Like the bathroom issue. Rather than having 100+ bathrooms just for each gender, making bathrooms gender-neutral would just change the signs from "men and women" to "urinal and no urinal" or just make one big unisex restroom.
If you're still confused, I suggest you try this video: https://www.youtube.com/watch?v=8kX5vNmat7Q
I myself identify as neither male nor female, and I am perfectly comfortable with answering any of your questions. Please try to be respectful, though. Cheers!