Does it really function all that well for conveying a neutral gender? No.
But, then, nothing in English- or the vast majority of languages out there- does. 'He' is existent and valid and melds seamlessly into the language structure, which is better than anything else can claim under most circumstances. India's Hijra- the best we've got to work with as far as nonbinary gender systems go- falls the fuck apart under even the slightest of pressure.
Is it appropriate for gender-neutral people that don't feel comfortable identifying as any gender at all? Hell the fuck no.
If he's gender neutral, then feeling uncomfortable about either gender pronoun is generally unlikely. Not having a gender identity means not
caring. Kinda in the same camp as asexuals. I'll grant you, it's pretty rare to identify people based on their sexual preference, so the analogy falls apart when it comes to pronouns... but, still... a lack of emotional or psychological association with either (any) gender makes it pretty simple to just not worry about it.
And they can always borrow from other languages. Hijra is such a pretty sounding word. And even in India, there's male and female pronouns... well, male and female suffixes that function much like pronouns... "a" for males, generally... and "i" for females. A structure that applies even to first person. I'm by no means an expert in the language, but like most suffix heavy languages (Spanish is another) you can convey the same details as a full (if short) English sentence. Gender and activity all at once (he is working, she's going to dance). As long as that word's a verb. Granted, a native Hindi speaker will grammar Nazi the fuck out of you for it, but it can be understood.
It's the gender
other that gets to be problematic. The ones that claim that they are neither male, nor female, nor some balance between the two (and nongendered IS a balance between). And since this is the personality bubble that tends to include people who believe they're really the reincarnations of dragons... well... that's a whole other conversation entirely...
What pronoun would you suggest to use for such a person?
Honestly? As someone who actually qualifies as gender neutral (or fluid, specifically). The people who imagine the word matters really need to lighten up. Gendered words- or for that matter the very concept of gender- are inelegant, sloppy, and more often wrong than right even for the most baseline of the cisgendered. It is quite literally impossible to have a gendered word that's truly accurate for even a single member of our species. There's only "close enough" and "not quite close enough".
But being all confrontational and in your face about it is just annoying and self absorbed as fuck. People like that make people like me ashamed to be any kind of gender queer. Much like most homosexuals feel about campy gays. They're a painful stereotype that perpetuates painful stereotypes, and most people who are the real deal wish they would stop embarrassing the rest of us.
They (not singular they, plural) could easily just pick what they prefer, or pick none and let those people use whatever they think fits better. Either way, it's best just to accept that most people honestly don't give a rat's ass one way or the other.
The most logical solution, honestly, would be to eliminate gendered words entirely. Pronoun or otherwise. It's well and utterly
impossible, but it's the most logical solution.
Do you simply refer to them as "that person"?
I tend to default to closest standing personality relation to gender. Sometimes, for the sake of edumucation, I'll let the people I'm talking to know that the 'he' or 'she' in this case is nongendered. But that's rarely something that comes up. And for the transgendered, I just refer to them as their preferred option.