A way I could see it working.
If you are playing in "Single civ gameplay mode"
Your civs UA, UI, UU and Unique civics Tree are locked to its native age.
However, its Attributes are always active... so in Antiquity, America/Mughal/Inca (since they are out of their Native Age) will all be able to
have 2 "branches" in their Antiquity Unique Civics tree
the "Antiquity Economics" Branch
and
the "Antiquity Expansionist" Branch
Mongols would also have the same "Antiquity Exploration" civics available but they would have the "Antiquity Militaristic" civics instead of the "Antiquity Economic" ones.
That way there is some uniqueness, but the devs just need to come up with
2 civics per branch
6 branches
3 ages
36 civics and associated traditions
and civs would still get the appropriate Attribute points.
That way
1. Tradition Abilities aren't unbalanced
2. America and Russia play differently in Antiquity
3. Not a lot of extra work
4. And it can be a game mode so it isn't imbalanced (ie the AI has to do it to)
You could add on that any leader can only choose a civ from ones they unlock (or an Antiquity one they are associated with if they don't unlock any because they are an Antiquity leader)
[with a Duplicates allowed/Avoid Duplicates rule option]... it might be fun to see Ben's America v. Harriet's America first come into contact in Exploration
Funny you should mention America and Antiquity in the same line . . .
I started a 'thought experiment' on Antiquity/Exploration America simply because America (British-derived Atlantic coastal settlement North America, to be exact) is the most quintessentially Modern Age Civ in the game, since it had no separate cultural, ethnic, or technical background before the Modern Age - geographically the area was occupied by people of different aspects in all of those areas and their own background was Distant Lands.
I think the way to extend such One Age Civs into other Ages is by de-coupling their Uniques (which, after all, in America's case would be virtually all Modern Age) from the Age.
That is, an American Unique might be a civilian Pioneer unit, representing the colonists that pushed onto the backwoods of eastern America in search of new land almost as soon as they landed. But, this unit and its attributes do not have to be Modern - they could just as easily push into New Lands in 3000 BCE as in 1750 CE so the unit could just as easily be an Antiquity Unique as a Modern one.
America in Civ VII right now is Economic Expansionist. This is the iconic basis for the Civ, and it should keep these attributes, but express them differently, in All Ages. After all, if there is a defining characteristic of the Americans, it is that we were land hungry and spread out as fast as possible for as long as there was anyplace to spread to, and that for most of our history we worked our collective tails off building things and trying to get rich.
So, Antiquity Age America: Economic Expansionist.
Unique (Civilian) Unit:
The Pioneer, a Colonist that can move like a Scout, disregarding most terrain, and has a defensive (cannot initiate combat) factor equal to a Warrior.
Unique (Military) Unit:
Colonial Militia, a Warrior automatically generated in every Settlement as soon as it is founded.
Exploration Age America: Economic Expansionist.
Unique (Civilian) Unit:
The Yankee Trader, a Merchant that produces more Gold from trade routes
OR
Pioneer becomes a Homesteader, a Colonist that starts a settlement with extra population.
Unique (Military) Unit:
Rangers, a melee infantry unit that moves like a Scout.
Modern Age America: Economic Expansionist.
Unique (Civilian) Unit:
Prospector - as now
Unique (Military) Unit:
Marine - as now
OR
the Ranger becomes a Modern Age unit, as Rifle Infantry with Scout movement.
The two things that I think that could be used to provide continuity between Ages for the Continuous Civ are the use of the same Attributes, and the potential to have Units that 'traverse' the Age Change into the New Age with new factors but similar attributes, like the Rangers or the Pioneer/Homesteader.
After all, if the continuous Civ doesn't have some automatic continuity among the Ages, why bother with it in the first place? Changing most of the aspects of your Civ between Ages, after all, is what the game does now - no need to do anything if we are just going to continue the same system under new names.