An "American" civ in 4000 BC clearly isn't historical, but (IMHO) isn't impossible or illogical, either.
I think of the first game as basically happening in a divergent universe (timeline splitting in 4000 BC), on an alternate Earth where the continents are configured differently. (Unless playing on an Earth map).
So, the Atlantic could have been super narrow or non-existent, and the Angles, Saxons and Jutes could have just gone straight to North America (skipping Britain) and migrating much earlier than they did.
On the flip side - why couldn't Romans survive to launch rockets to space? There were an "India" and a "China" contemporary to Rome, end they did end up launching rockets to space. Obviously, it would be a much different "modern day Rome" with likely no columnated porticos etc and legionnaires in sandals, but you get my drift.
It is alternate history but nothing causes cognitive dissonance.
Compare that to Civ VII (I am on my second playthrough now), in both games I don't know who the hell I am playing against and which leader leads which civilization. The game doesn't make this very prominent anyway, I have to hover over banners in the upper right corner to remind myself who is who.
My first game had Jose Rizal, I think, leading, if I remember correctly, Hawaii, then Majapahit, and finally Japan; places thousands of miles away from each other in RL Earth and not very much culturally connected, either.
My current game has Ben Franklin leading Ming, and Lafayette leading the Normans. These can very well evolve into Qing and the Americans, and then when Ben Franklin pops up on the screen, I will always initially presume I'm talking to the Americans.
Huge cognitive dissonance.
(On a separate point - I wish they simply took this opportunity to get rid of immortal leaders. I think the first game had them as a joke; Sid liked humor in his games, and it was funny to see Lincoln in Stone Age clothes and Caesar in a suit I guess. But the current Civs are dead serious and they still have immortal leaders).
Anyway, they made the game the way they did, and I accept that. My only wish is that when there are sufficient numbers of civs and leaders, an option will be added to only allow historical transitions. This won't take anything away from the players who like this approach, but it will mean a lot to players like me.
The thing is, if you follow your explanation why an America in 4000 BC exists, you could do the same for the switching. The explanation for Egypt-Mongols-Japan seems equally logical (or even more – not that it matters). The Mongols almost reached Egypt in their conquest. If they would have just concentrated a bit more to the south (and didn't lose their first battle against the Mamluks of Egypt in what is modern Israel), it seems plausible that they conquered Egypt and either kept it as part of their Empire (if it didn't fall) or established a Khaganate of Egypt. So Egypt > Mongols is checked if we allow a few "what ifs" (and we don't even have to bend the world to make Egypt more Eastern than Persia or something like that or have things happens thousands of years earlier). Now, the Mongols tried to conquer Japan in history, but failed. Yet, if they succeeded, they might have blended into the Japanese (as a nobility, for example, as is often the case). So, Mongols > Japanese is also not really implausible.
I don't understand why people keep complaining about Rome not having the right options. If you keep to layers, and go to places like Rouen, you can clearly see Rome > Normans > French Empire. And in Spain you can clearly see Rome > Spain (e.g., in Cordoba), but I actually don't know examples where you find the French Empire on top in Spain itself. You can find these three in Southern Italy though, for example. In contrast, the often heralded India path has no such examples that I'm aware of. No place saw Maurya > Chola > Mughals afaik, and there is also not really much going for the line except that they are all part of the modern state of India (mostly, the Mughal heartland as much in modern Pakistan). The India path to me looks like Rome > Norway > Turkey tbh, and not like Rome > Normandy > Britain.
I know it's tempting to think of Rome > Italy as the only correct path, but this is just a very narrow (and misleading) geographic proximity rule imho. Sure, Rome's heartland has been the Apennine Peninsula for a long time, but the Roman Empire thrived in many places, and people and ideas migrated also a lot inside the Empire. Capitals changed or were split (to e.g., Thessaloniki, Trier, Antiochia), the Emperors (which were also from almost any place of the Empire, really, not just Italians, see Trajan or Constantine for example) moved away to Split for example, many of the most prestigious building work is built far away from Italy. And finally, we have Constantinople as new heartland (with Ravenna taking a Western spot at some point), and over time people speak Greek as main language etc. So, historically, the continuation is Rome > Rome (= Byzanz). Any of the Italian states and cities might have geographic proximity to Rome (city) on their side, but culturally (and ethnically) the continuation isn't larger there compared to e.g., Spain or Southern France. They all kept many Roman systems, part of the local population stayed, but other parts were exchanged (the Lombards are usually the prime example here in Italy itself). The HRE has obviously a big claim as alternative to Rome, as they were an alternative historically, taking over as simultaneous Emperors and ruling large parts of Italy (and various other important parts of the Roman Empire). I would actually like very much to have one of Tuscany, Genoa, Venice (without the 1 city limit), Papal States, or Milan as civs at some point, but I don't think that they are really a 'better' continuation than Spain or Normandy, either looking from past (= Roman Empire), where they are just a random geographic splinter of the Empire, nor looking backwards from today, where Florence isn't more Roman than York.
I honestly think that a lot of this has to do with familiarity. What we are used to doesn't cause as much cognitive dissonance as what we aren't used to. What we know and have accepted as "this is just how civ works" 20+ years ago is deeply engraved as normal and plausible. What challenges this fictional world in which civ takes place that we've created in our minds is automatically seen as wrong. But it really isn't more wrong than what we are used to. Getting used to Ben Franklin being Ben Franklin, and not the leader of America might take some time and games, and might be confusing for a longer period (it isn't for me, for some reason though). But it shouldn't be more confusing than Montezuma ruling his peaceful island empire, being the spearhead of technology, having a great influx of tourists from Russia and Australia, a large cavalry army, and owning some Arabian cities. It's just that we learned to accept that the game allows Monty to have an empire that has nothing to do with the one he ruled historically – we don't even expect any more that he behaves historically or rules an empire that resembles the Aztecs. In contrast, we (or at least many of us) haven't yet made the disconnect between leader and civs in our minds. As you describe, you see Ben and think of America. Maybe, in some time, you see Ben and think science, getting more endeavors – and aside from America (which he often ends up leading) of him going from whatever he currently is (Greece, Rome, mostly) into Normans if he can.
All that said, two last things:
a) I hope for more civs to smooth out the paths and give more options. I think with 10-20 additional civs, it will feel much better to most players – even to me as someone who doesn't care much about historical or geographical paths in many (but not all) of my games. And I honestly think this is a point most can agree on: more civs will improve their experience. Yet, as it seems, it will come at a hefty price.
b) As said above, I don't think history is too important for how the game plays out and what it does. I don't see it as alternative history, but as a tournament between leaders duking out who is the best in this round. An embellished and historically themed Super Smash Brothers, if you want.