Yours seems like the most sane idea, actually I’ve never understood why we have separate bathrooms anyway it’s not like two people go in one stall together. It’s all predicated on the principle that people will behave morally when around strangers in public. We’ve kinda lost that already.
Although I still like having men’s only bathrooms because this way we don’t have the long lines women always seem to have. The urinal really is an efficient use of space.
In a public setting, just make sure there are locking stalls for privacy. There can still be urinals for men who want to get in and out quickly.
Way too much hysteria over public bathrooms. We humans have very similar bodily functions. I've been using public bathrooms for 50 years and never had an issue. Should it bother me that a woman is in the next stall doing her thing? Not at all. Nor should it be a problem vice versa.