3 places where I feel like gender separation doesn't really make sense
Sports
Separation of men and women in sports is fairly admirable as it gives people a chance to showcase their skill that would otherwise be outcompeted. It additionally is nice as women are a group that are often discriminated against and exposure in previously limited areas is nice. However, I don't think that a strict gender separation is really necessary. I think that an ideal system would allow anyone with higher skill to go to the top of their league, relative to the physical ability determined by their genetics. I'm not very into sports, but I get the impression that people's enjoyment often comes more from people's character and effort than the absolute magnitude of their ability. Short v.s. tall people in basketball are one example that comes to mind; a shorter person would require much more skill to reach the same level as a taller person currently. I'm not much of a wrestling person, but I think this is addressed there through weight classes. A possible wider idea is be some meta-classification into classes based on the characteristics that cannot be changed with more practice or other self-improvement.
Bathrooms
this is basically a summary of this very silly 2kliksphilip video
Urinals are more space-efficient than toilets, but typically only are found in men's restrooms. Therefore, with equally sized men's and women's restrooms, the men's restroom gets higher throughput assuming an equally sized demand, and has under-occupied stalls compared to the women's room. Even if both are perfectly sized for average demand, there will still be inefficiencies when outlier groups come in. There's really no reason other than tradition to not just separate out the urinals (if desired) and unify all of the stalls, with full height walls if you think it isn't private enough (Really, as a 6 foot 2 guy, it's silly how low stall wall tops usually are).
Pronouns
I was working on a thing recently and had to refer back to someone in a sentence that already included 2 men. I know that some other languages have primarily gender neutral pronouns, but a concern that I have had is that it would make it harder to tell who someone is talking about. I think there are some alternate systems that are better at resolving general ambiguity though, like having different pronouns for the person most recently named vs. 2nd most recently named, etc. There might be languages that do this already, idk, I just speak English and a teeny bit of Spanish lol. I haven't put all that much thought into this but I'm sure there are ways that could make this problem in communication even easier than it is currently. If we could ever get an opportunity to modify common speech.
In response to your section on pronouns:
As you identify, it's useful to have multiple different pronoun sets to refer to different people to reduce ambiguity when speaking and writing.
We could hypothetically base these categories on anything: we could have one set pronouns each for men and women, but we could also set that dividing line somewhere else. Maybe we use one set of pronouns for family and a different set for non-family. Maybe the dividing line is rich/poor. Dog person / cat person. Personality type. Horoscope. Favorite color. Color they're currently wearing. How recently they entered the conversation.
Some of these sound pretty reasonable and others sound really useless. A gender-based pronoun has problems, but it's useful in that it's often a useful differentiator between any two random people. This wouldn't be the case for a hypothetical rich/poor pronoun system.
Now that I think of it, a Chinese zodiac calendar-based pronoun system would be really cool. The 60-year sexagenary cycle would give us 60 different pronoun sets for each year people are born, allowing each pronoun (fire dog, metal rat, etc.) to gain their own associations over time, though constantly changing as people from each cohort get older. Because people don't really live more than 120 years, you would also only ever have two generations of each birth year. So there would be an "elder fire dog" and "junior fire dog" and it could be so interesting and artistic and poetic with two very different groups sharing a common pronoun... So much room for symbolism and reflection 🤤
But anyway, snap back to reality. Neo-pronouns already exist in colloquial English, especially online. I'm not talking about xe/xim, I'm talking about bestie, oomfie, anon, homie, my guy, my brother in Christ, girlypop, etc.
We should recognize these for what they are (pronouns) and normalize their use. They all have different contexts, connotations, and use-cases, but they are absolutely usable sets of pronouns. Some of them are still gender-specific, but the important thing is that gender is no longer the primary relevant factor in pronoun selection. Let's have 100 different pronouns, and everybody can use any of them depending on the context. It would be awesome.
I love your take on pronouns. If we used western astrology concepts we could go full Battlestar Galactica. A lot of people don't realize how elective western astrology actually is, especially when you bring up how much of modern soft / pop psychology is based on it. The Meyers Briggs correlates directly to the four humors / classical elements.