This is such a great game, I feel bad to talk about two things that are holding it back from near perfection: 

1.  Almost no reason to transcend eras
2. Random cultures pop up where they don't make sense.

When the Mycenaeans become Mayans who become English on one side of the world while the Olmec-Celtic-Byzantines are on the other and have no contact, it breaks the absolutely wonderful aesthetic and narrative this game provides.

One solution could be when you have enough stars to move to the next era, you will spend your accumulated score points to choose your next culture.

The culture will vary in price depending on your previous choices as well as the world around you. Your score will still matter as the end score victory will be based on points earned, not available to spend.

Settling as the Phoenicians, your people look towards the classical era at the many opportunities. The cheapest option would be to transcend, while another affordable option would be to invest in war elephants as the Carthaginians. Another economical option has been provided through your relationship with the Zhou. Trade and your diplomatic treaties have brought the price for the Han almost as low as the Carthaginians, and the Diplomatic affinity will help you with your geopolitical situation. So, Han it is.

Turns later, the Medieval Era is about to dawn, and opportunity is knocking. Through your political maneuvering, you have provided your people an era of security and prosperity, earning stars in every category. A successful campaign against the Romans boosted your cities as coordinated attacks from you and your allies resulted in maximized gains with minimal loss. With your main enemy gone, you find your civilization to be on top of the world. Unfortunately, you find yourself bored. You wish to test yourself against the strength of your allies. As you look through the choices for your next culture, the Aztecs have your attention. You have not been in contact with any related cultures so the price is pretty high, but the victories of the previous era have provided enough score to make the transition. So, Aztecs it is.

This approach would allow civilizations who were on top of the world to act as cultural leaders, impacting the choices of the other empires around them, while those who are underdogs would have to weigh their options and modify strategy to get the culture they desire.

The world would have a more organic look as cultures would evolve based on their environment.

Interesting! Some filiation between cultures and culture legacies would improve narrative and immersion a lot indeed.