There's not much in the 'standard human' pallette (Hyur/Wildwood/Sunseeker) but Lalafells seem to have more of a yellow range in their options. I'm not sure why they made that different.
However, the "lack of Asian faces in a Japanese game" suggests that whatever the source of the issue, it's not about "failing to represent other races because they are foreign". It's also a fantasy world that is (at least initially) set in a Europe-equivalent, so it does make sense for the characters to look more European - not that the 'Asian' Hyurans look any different once we get there.
For what it's worth, there doesn't seem to be an Africa equivalent, at least geographically - unless it's supposed to be Meracydia, but it's not connected by land to the other continents. (Also I've seen people say that Eorzea itself is meant to be Africa, based on the approximate shape of the continent, but that doesn't make sense culturally. Othard isn't shaped like Asia either.)
It's not Earth, so how can we know? Maybe their origin doesn't mimic human origins on Earth. Again, there isn't an Africa for them to migrate out of, at least by a land route.
The lorebook information on Hyurans seems to say they came to Eorzea out of Ilsabard, and they're found across the Three Continents and the New World. (Again, information is lacking about Meracydia so they could be there as well, but we'll probably get to that sometime later.)
You said "red, grey and green people". Not Hyurans. The other races are still equally people. Hyurans' defining trait is being actually-kinda-boring and not venturing outside of (a subset of) real human traits. You pick a different race if you want unusual skin colours.
If you want to talk about lack of variety overall, you still need to look at all the options - otherwise you might be saying "I can't make my Hyur look like X" when actually you could have X if you used a different race. It depends which elements of the appearance are most important to you.
Limsa is in the southern part of the map, but that's not the southern hemisphere - the map just shows northern-hemisphere continents. (Additionally, Sea Wolves originate from further north, and seem to be the Eorzean equivalent of Vikings.)
From what I remember of geography class, there's roughly a band of tropical climate around the equator, then desert either side, then temperate climate. Eorzea seems to be temperate with desert only in the southern region, plus Gyr Abania. (I wonder if the map is perfectly 'straight' or if there's some distortion going on. Or you can just blame it on aetherial states messing up geography or something, like Coerthas's eternal winter.)
But yes, skin tone does seem to be somewhat arbitrary - although despite having the option for Duskwights to have darker skintones, the few we actually see tend to be pale grey (eg. the researchers at Issom-Har in the South Shroud) or greyish-tan (possibly those that have been out in the sunlight more!). I can only think of seeing one really dark grey NPC, a Wood Wailer standing guard that progresses a quest when you speak to him, and isn't seen again after that.
For weirdness elsewhere, there's the few dark-skinned Ishgardian Elezen when most of the population are pale. It seems to be an individual/family trait rather than race-related.