For all people saying this or that is "immersion breaking", I wonder what are you criteria to find that immersion breaking while having Rome and Byzantium (which are the same civs, heck, they called themselves by the same name) or Byzantium and the Ottomans (who both exactly had the same city as their capital, but apparently that doesn't breaks immersion, perhaps people don't have object permanence and changing Constantinople into Istanbul is enough to say that it's two different things). But that's not the topic.
On the poll, I took option 1 which is objectively the best choice gameplay-wise if you want to change core elements of your faction during the game. I already explained why at other places, but let me try again (as each time I refine my thought).
First, with the question of immersion: people think of their civilization as something continuous while leaders can come and go. Except that having a leader "dying" at each age is just as nonsensical as having them for all the game. Either way their near-immortal people leading your civilization. Having them "die" at one point would be odder than them living during an entire era, in my opinion.
But the main argument for eternal leader and fleeting civs in gameplay related. The main issue that a lot of people sensibly rose with this whole system is the potential lack of identification and continuity with your civilization. "Oh, I was the romans and now we're mongolians" => sure, that might be a problem. But what is the main way you interact with other civilizations? With the civilization itself? No, it's with the leader. Each time you interact with another civ, it'll be the leader that you see, because the leader is more characterful by their very nature of being characters, that's nearly a tautology as this point.
If you want to feel a better continuity with a foreign civ, you need continuity in their flagship, in the main thing you interact with, you see in it. And, as for now and since at least Civ IV (never played before so I can't really tell), this flagship is the leader. It's their head you see, and their voice you hear since Civ V. Therefore, keeping a continuity has to go with the leader, it's in the way of things.
I returned to Millennia recently, and they don't have leaders, merely civilizations, and truth be told, I barely think of my neighbors or opponents by their culture. When I get a message "South Korea is at war with India", I just ask myself: who are those guys? Oh, never mind. While if I had a message in Civ VII saying to me: "Augustus declared war to Hatchepsut", I'd quickly know that it's the twinkish boy with stick legs who declared war on the fiery queen with no hat at all. Those leaders are characters, and we humans are designed to recognize faces and get attached to them, so we feel a better connection with the leaders than with the civ.
Elements to further my point: in caricatures in newspapers, countries are often represented by characters. We all know Uncle Sam, we French have Marianne with her phrygian hat, Germans were automatically recognizable when you saw a moustached man with a pointy helmet, and everyone and everything goes through personification. So Civ VII is taking the sensible road and having those personifications representing the civilization, despite internal changes. The French might have gone through more political forms of governments than a trostkyist assembly under acide, but Marianne was able to keep a certain continuity since the French Revolution; less so during the monarchic phases, but even there, you knew only France could be represented that way. Despite changes, despite evolutions, despite everything, now we have our phrygian hat lady representing us. Caricatures are a very potent way of depicting a civ, and that's what Firaxis is doing, because, while people might like it or not, it's objectively the most efficient way to quickly have a visual image of the country, sometimes even more easily than with the flag.
Other element to further my point: imagine you're playing a Civ VI game, and you have Chandragupta against you. Then suddenly, it's Gandhi. Two questions: 1) would you feel as if you play against the same civilization? 2) would you react the same way? Other scenario: you start a game, and one neighbour is Chandragupta, the other is Gandhi. Would you react the same way against those two neighbours? If you think of an opponent first in terms of civilization then in term of leader, then your logical reactions would be to react the same against both of them; but I dare advance that each player would react differently. Why? Because each leader is much more impactful in the way a civ is played than the civ itself.
Because if we take another situation, it's even more jarring: Eleanor. If you are in a game against both Eleanors, you (at least I) would react quite similarly. You'd now that you'd have to take an eye on loyalty as she might try to gobble up your cities with her artworks. No matter if she's keener on wonders or harbours: in both cases, the loyalty mechanic will be much more impactful and you'd play against FrEleanor quite the same way than against EnglEanor; at least, in a much more similar way than if you played against Chandragupta or Gandhi.
And if we compare with the most comparable element, we have Humankind, which utterly failed at it. Avatars in Humankind are some of the blandest elements of the game (and it's not a small proclamation for this game). Therefore, you had factions that constantly changed cultures, but you didn't even have a characterful memorable avater to keep continuity, which is was it was quite badly implemented in Humankind, something that Firaxis seemed to have taken into consideration.
That's why I think option 1 is the best way to go. You might personally prefer or think that changing leaders would be better -and it's a perfectly valid opinion, that I cannot deny and it's as legitimate as any of the other options-, but from a gameplay perspective, if you start from the unchangeable and unalterable axiom that you have to change the leader, the civ or both across the ages, the best choice to maintain a sense of continuity in the game is to have the leaders being the continuous element.