Start in the ancient era with 2 policy slots for your government.
Use culture points earnt during the current (example ancient) era to unlock more policy slots for the next (Classical) era. For example for every 100 culture points earnt, you get 1 policy slot for your government for the next (Classical) era. This does not stack with slots already earned from previous eras (For the classical era, if you 'advance' with 101 culture points for the entire classical era you will have only 1 slot)
To progress into the next era, you "click up" yourself, with like a "societal reform" button. This is when you will be able to use the slots you have earned in the previous era.
The culture cost of unlocking slots varies based on era/empire size etc.
You cannot progress through eras with technology. Instead each era has many (like 40 ?) technologies per era with the expectation you will not research all of them. There is however (strong) Tech diffusion, but it only applies to civs in the same era.
So you can advance more quickly but with less bonuses for the next era. Tactically, it could be smarter to delay era advancement to unlock more policy cards and slingshot past those that advanced before you. By delaying, you also get the tech diffusion boost towards anything that the 'more advanced' factions have researched, but they don't get tech diffusion towards anything you researched from the previous era while they were 'ahead' (For example, you might go through 8 technologies and unlock ocean going vessels in the ancient era BUT an opponent that advanced eras earlier than you will have to research a similar technology later in the game to unlock that ability).
Optional stuff (That I'm on the fence about myself)
You can only change policies at the start of a new era (after hitting the reform/age up button).
When you reform you pick a government and that determines the pool of policies you can implement
Replace leader abilities with leader favourite policy mechanic. When you play as a certain leader if you implement that leaders favourite policy you get double the benefit from that policy and extra yields from trade routes with other factions with that policy.
The 'Ages' are an artificial game construct to begin with: the whole world never went into the same 'age' at the same time, no matter how you define the ages/eras. Therefore, making more artificial restrictions on age progression is just a recipe for frustrating the gamer: giving them more artificial hoops to jump through and artificial 'progress' to an artificial goal.
So, rather than say 'you cannot progress through eras with technology" say that Each Era you are aiming for will require different paths and accomplishments. It is inane to have an Iron or Bronze Age without having the technology of bronze or iron working. Equally, it would require a convoluted bit of logic to require some Technology to reach an Age of Philosophy, Faith, or Enlightenment. None of those are, so to speak, "technology driven".
Rather than give the (human) gamer some tactical advantage to manipulating the Era progression, make each Era also come with a certain amount of Upheaval: new Faiths, Technologies, Philosophies are not comfortable for those going through a re-arrangement of their industry, economy, church and state, and that discomfort is frequently expressed violently or with some other negatives towards their leadership that got them into This Mess - even if the leadership had little to do with it. In other words, every transition in Eras/Ages should come with a Social/Civic/"Happiness" Cost. Likewise, many technologies should come with similar Costs, because, as examples, an Iron Age is going to upset all the bronze and brass workers, an Industrial Age will change the entire set of skills required in the workforce, and new philosophies/faiths can be guaranteed to upset everyone satisfied with the old philosophies/faith.
This base concept ties in with some of your thoughts:
Governments don't change voluntarily, as a rule: change of government, then, would be made necessary by the disruption caused by adopting new Technologies, Civics, Faiths, Philosophies, and especially by engaging a new Era which embodies several of those.
Policies/Civics should be, in many cases, specific to certain types of governments: expecting a civic of Universal Suffrage under an Absolute Monarchy requires 1984-type Doublespeak (or the 1937 Stalinist Soviet Constitution, an extremely liberal document that was, of course, utterly ignored by Stalin and his government!)
Leader abilities have become a Core Part of Civ, but leaders can also have a preferred Civic/Policy (or several) that gives them extra goodies. A goal, perhaps, should be that every Leader in the game could have a Unique Unit, Building, Improvement, District or a bonus to special circumstances regarding those (perhaps 2 - 3 out of the total of possibilities?) and certain Wonders, and a Favorite Policy/Civic. A government that combines a Unique (or more than one) and a Favorite for the same Leader would be especially powerful for that Leader - like a Royal Palace unique Wonderous Improvement and Civic of Central Control all under an Absolute Monarchy type of government - Louis XIV, the French government as organized by Richelieu, and Versailles anyone?