I don't speak French, but I'm a native Spanish speaker and I have two observations:

First, it's true that Romance languages use gender in this manner. I was just pointing out that the specific example chosen didn't show this. The example has nothing to do with the gender of the speaker, but with the gender of the object under discussion; it's similar to you saying "she" or "her" when referring to a woman; it enables me to make no assumptions about your own gender.

Second:

> Anything the speaker would say about themselves would have to be gendered accordingly.

In Spanish there are careful ways to avoid making assertions about yourself in a gendered manner. They are a bit more roundabout, but can be done. My point is not that this is the most natural way of speaking, just pointing out that it's technically not true that you must always make gendered assertions about yourself.

Example:

Instead of saying "I'm very tired", "estoy muy cansado" (gendered male language) I can say "tiredness overcomes me", "el cansancio me invade", hiding my gender. Yes, it's a bit artificial but not so uncommon I have never read it. So in Spanish you can make assertions about yourself without betraying your gender. Don't know about French or other Romance languages, but it wouldn't surprise me to know you can employ similar tricks. It's probably harder in Spanish than in English to hold a conversation longer than a couple of sentences while hiding your gender; it will start sounding very unnatural, I grant you this.

I guess you could do it to certain extents, but it’s likely to be considered incorrect French, or just really weird. Now if you’re writing a song or a poem, it’s more common to see that kind of construct, as a way to play with rhythm and sounds, so it’s not entirely impossible.