Anyone could fit into the position, really, it was never specified what race the Nerevarine would be, and I don't think what race Nerevar was makes much difference. Any race could potentially be the Incarnate, but I think a Dunmer would be a bit more likely to accept to take up the task of fulfilling the prophecies, since the Nerevarine would supposedly drive outlanders from Morrowind, though that is not something you actually need to do in the game, you do not know that at first, and it seems a bit illogical that anyone of a race that is not Dunmer would try to fulfill such a prophecy, being as that would involve driving your own kind from Morrowind, though if you look at it that way, it wouldn't make much sense for ANYONE who can do it to try to actually try to fulfill the prophecy, since the Nerevarine must be an outlander, and even outland Dunmer probably would not want to drive all outlanders from Morrowind, since they themselves are outlanders. In the end though, Bethesda left the prophecy pretty open as to what race the Nerevarine can be, obviously done to make it work no matter what race you choose.
Ultimately, that aspect of the main quest never made much sense to me, the obvious advantages of having a native hero in your pocket aside, I'd think that even with the Emperor's orders, the Blades would at least question trying to fulfill prophecies that the natives claim will lead to the Empire being driven from the land.
On Vvardenfell, everyone not born on the island or raised elsewhere is an "outlander". Even Dunmer from the mainland.
Indeed, and as long as people could tell that you were not a native (which would probably show in your accent and ways, even if this is not reflected in the game.) the whole thing would not apply.
And besides, I recall the thing about the Nerevarine being an Outlander
It should be pointed out that all of the failed incarnates were Dunmer.
That's natural, since aside from you who were sent to Morrowind because the Emperor seems to believe you might be the Nerevarine, the only people who could really WANT to see the prophecy fulfilled are Dunmer, it has an obvious appeal to many Dunmer, especially Ashlanders, who would likely want nothing more than to drive out the foreign occupation and cast down the false gods of the Tribunal.
There's also the fact that they were FAILED incarnates, aside from getting killed while attempting to fulfill the prophecies, a good reason why they might fail is because they don't meet all the requirements, I'd hardly say that the races of people who couldn't fulfill the prophecies say much about who the one who can fulfill them will be.