I don’t agree with separate eras…but the Crises could be more significant to help lessen the impact of the transition itself.
This pretty much sums up the problems with the Crisis periods now: they are annoying, but generally not a crisis. I have never lost a settlement to a crisis, and rarely even seen AI Civs lose a city or settlement. What kind of Crisis: invasion, plague, general Happiness deficit - seems to be utterly random, as are the actions the gamer can take to alleviate the crisis: a miscellaneous combination of repairing destruction by mobs, adopting Civics, keeping general Happiness up.
I think that making the Crisis both more specific and more dangerous, and giving the gamer/AI more specific ways of dealing with the Crisis which in turn lead to specific outcomes at the end of the Age would do two important things:
1. It would make the Crisis period a genuine Age Ending period - you would be forced to deal directly with the Crisis instead of riding through it while continuing your regular activities.
2. One of the specific outcomes from your reaction to the Crisis could be one that allows you to keep the 'same' Civ in the next Age. I put same in quotes because, of course, it would not be completely the same: among your reactions to the crisis would be some that would change your Civ in some pretty fundamental ways, like the differences between western Imperial Rome and Byzantium or between the Han Dynasty and the Sui or Tang Dynasties.
To start the discussion, note that the Crisis period that ended western Rome and is generally considered the model for the Antiquity Crisis had the following actual causes/events and effects:
1. Plagues that depopulated Rome, which lost an estimated 1/3 of its population in the 2nd and 3rd centuries CE. That in turn left large parts of the Empire depopulated, so that German peoples were invited in to 'refill' those lands. But those people came with their own leaders and culture, which destabilized the Empire.
2. A tax system so inefficient that Rome struggled to pay for a 500,000+ man army, especially when large numbers of tax-payers went missing from the plagues. That in turn meant more and more of the army was in act recruited from German and other non-Roman peoples. They were very good soldiers, but, again, not such good Romans, and the increasing reliance on them, again, destabilized the empire.
3. A set of loyalties that did not center on the Emperor or the Imperium, but on individual patronage - from and by army commanders, governors and such which made it easy for those to set up independent Power Bases and try for the Imperial throne. That resulted in the 'crisis of the third century' in which Rome spent most of that century changing Emperors forcefully and to the accompaniment of civil war, assassination, and general disruption.
So, any Crisis potentially giving the same amount of disruption - that wipes out a Civ completely and replaces it politically, militarily, and eventually culturally - would pretty much have to have similar characteristics:
1. Dramatic loss of population, either directly by dying of plague or indirectly by moving out of Unlivable Cities to the 'safer' countryside.
2. Loss of Loyalty. Settlements, areas, military units - none should be entirely trustworthy and rebellion of some or most a near-constant threat.
3. Loss of territory and settlements. Western Rome famously had already lost control of Britain and most of Gaul/France before Rome itself was ever directly threatened, and wasn't even trying to get them back. By the time Rome 'fell' in the late 5th century, Britain, Gaul, Spain, and north Africa were forming entirely new Post-Roman states. In game, that means either new (Exploration Age) Civs or IPs.
Such wide-spread dismantling of the Civ you just spent 100 or so turns building would be a very Hard Sell to gamers, but the negatives could be countered by giving the gamer mechanisms only available in the Crisis to start forming their new Civ - either a continuation the old one or an entirely new one, or something in between - the differences, say, in an Antiquity Roman context between the Byzantine version of 'SPQR', the post-Roman Abbasids of the middle east or Al-Andulus of Spain, or the German/Frankish Merovingian Kingdom of Francia.