Thesis: Each has a unique "look". While races do play a part, the role the effect they have is minimal in an Elder Scrolls setting is minimal because all of the races have basically humanoid builds, and couple additionally have tails. The true difference maker is not the race of the opponent, but rather the enemies' gear.

To use an example, when I played Skyrim the first time and I was adventuring in Morthal, after getting all the quests available in town, I cleared out all of the quests outside of town and worked my way back in. During this process, at least a half a dozen or more times I fought against the Forsworn. After all those hours of killing Forsworn I did not even realize that the Forsworn were Bretons until I was locked up in a mining jail and one of the characters locked up with me told me they were. I'd just assumed the the Forsworn were Nords since most people in Skyrim are Nords. Since I didn't know much about the Nord's either. However I could always tell the difference between a Forsworn and a member of any other faction simply by the crazy animal skin armor they were wearing.

To use another example, there are Nords fighting for both the Stormcloaks and the Empire, and you never confuse who was fighting for which side when you saw them because the armor they were wearing was so distinct that you could tell the difference immediately.

The same design principal can easily be applied here. Yes, the leadership of the factions can still be the representative governments of those particular races. Yes their motivations can be exactly same. But if you think about it from, purely for a logistical stand point and different areas these factions come from and their wildly differing climates, the gear that each of these factions are using should look different.

The main Armorers of the Aldmeri Dominion are the High Elves. The main Armorers of the Ebon Heart Pact are Nords. The main Armorers of Dagger Fall Covenant are Orcs. The types of gear these different races make, are distinct enough from one another, that even a casual observer would be able to see the difference.

To give a real world example, my brother lead a Armor Battalion during the first year of the Iraq war. Specifically he was the commanding officer of some 45 M1-A2 Abrams tanks. On one particular mission, there was a joint operation the US Military and the Iraqi police, which was a proto police/paramilitary force that the US army trained in Iraq during the first year of the invasion which has sense split off into separate organizations. His role in the mission was to secure the boarders. He knew at the time that bad guys had gotten their hands on Iraqi Police uniforms. So during the fight he gets a radio message from one of his tanks on the opposite side of town. There is was suspicious report of ten Iraqi Police heading in the same direction as another patrol that went by recent, and two of them in the back were armed with shoulder fired Rocket Propelled Grenades. Well my brother knew that the Iraqi Police didn't have RPGs, so even though insurgents were wearing same uniforms he knew right away that they were enemies.

I'm sorry but this way it would feel more like a civil war to me. I mean, it's just the races picking a side within Tamriel. They will even fight their own species. That is not really the point of this game. It's more about racial and factional allegiance and trying to take the throne for your faction. One great leader of each race are at the head of each faction. How would you fix this with your approach?
Also it would be kind of weird that if you were a Nord in the DC, that you couldn't enter skyrim. This factionlock and racelock are tied together.