Totally agree with your premises but kind of came to the opposite conclusion.
It's like clothes: girls wearing pants is more popular than boys wearing dresses, but that doesn't mean girls shouldn't wear pants; it means boys should wear dresses more often so that it doesn't look "weird" to people. It's also like popular bands, movies, etc. Just because it's popular doesn't mean you can't like it for itself. Masculinity is culturally valued more than femininity, but if you like masculine things for themselves (not just because they're popular), you shouldn't have to pretend you don't. There's no shame in genuinely liking something popular, as long as you like it independently of its popularity and also have no shame in liking something unpopular.
That said, I don't like
George on a girl at all! But I do think you should choose a name by the sound and overall feel. I also think some boy names (that have never been commonly used for girls) are feminine without being typical "girl names" and vice versa. When I think of "masculine" and "feminine", it's not just about whether the name is a "girls name" or a "boys name". Plenty of names are neither feminine nor masculine without being strictly unisex names.
(examples: I think
Astrid is masculine,
Silas is feminine, and
Conor is neither. But
Astrid is definitely a girls name while
Silas and
Conor are definitely boys names. Some names just have qualities that I percieve as masculine or feminine.)
In practice, neutral is better since you want the name you pick to fit your baby's personality and you don't know if they're particularly masculine or feminine when they're born. But just on principle, you can have any name without reading that much into it. Having a boy name doesn't mean you're trying to be a boy. You can be a girl with a "boy name" just like you can be a girl and wear "boy clothes" (and both of those things are different from a masculine "girl name" and masculine "girl clothes"). I can wear a regular T-shirt, not a "girly" one from the girls section of the store, without people saying I'm trying to be a boy. So why would a name be any different?