To simply a long winded several paragraphs the religion feature while interesting as a game mechanic fails to capture well the aspects of any religion except as a vaguely macro-economic system of public works. Religious experiences or religious leadership are occasionally featured or interacted with in a very abstract way through events or the civics. The holy sites could have been an alternative to wonders. Say for example an international airport or something like the Diolkos. Another option could have been ideology as opposed to faith. I really have to question the blatant gamification of the religions and found that while I could take advantage of the systems I usually ignore religion and used it passive aggressively mostly for stability and food. In both of the following suggestions you remove the holy sites.

These could be available as "great works" and they could effectively trigger an empire or population bonus - it could even be the other way around. I actually see another way forward where you change "faith" to something called "values" and these values trigger unlocks - the unlocks require a building but would be available to anyone with the same values. When you build the structure you get the bonus. Some of the buildings can have multiple subsequent improvements, a system of roads could lead to a system of freeways. Some would be a one time thing, a Dam for example. With this system I would propose two major changes. One the population unlock remains but the tier system is limited to the Era so you can't end up with too early a bonus but there's also another incentive to moving on to the next Era. The other being that it only sometimes generates a grievance - on if the accumulated values are in opposition to each other (food & stability vs science & industry vs money & inf) and only when your entire population has gone over to an opposing value system. Taking a middle of the road value set yet might be more advantageous. That system is a little harder to work out the details for on the fly because each value should have a corresponding great-work that needs to completed which grants you the 'tenant' bonus and potentially a subsequent effect or bonus or social ability.

I feel like there is also a way to keep Religion similar to how it is now but we'll be specific and call it Philosophy instead. Semantically the Tenets become the practices of the philosophy. But instead of faith well base success on the Philosophies merit. Now with that out of the way let's be a bit broader but also specific to our approach and lump some philosophy together in three macro blocks: Naturalistic, Hinduism, and, the 100 Schools of Thought. In this system we also remove Holy Sites. Instead merit increases primarily in the following ways.

Hinduism
+ 10 merit
+ 10 inf on your capital
+ 10 merit & + 5 inf on wonders

100 Schools of Thought
+ 5 merit
+ 10 inf on wonders
+ 1 merit on your EQ

Naturalistic
+ 10 merit
+ 5 merit on wonders
+ 3 inf / +3 merit on non capital city centers

--- Yea I don't know if there balanced either but their all pretty different from each other already.

Your ideology may limit you from embracing certain philosophies if you are on the extreme of an axis. I think that should mean that no one is restricted by another player choosing any of these choices. Grievances should also be changed to better reflect total much larger social divides when a nation is philosophically and ideologically opposed. So for example Purge Idleness is a collectivism philosophy - it doesn't change your society - but if your society's ideology is on the individualist extreme you will be prevented from choosing Purge Idleness. 

Tier 1 - Ascending out of the Neolithic with enough followers is a special case here - the requirement is 3 stars earned.) 

C   Purge Idleness                   +10 food (capital) 

A   Abstain From Intoxicants    +10 industry (capital) 

P   Seek Wisdom                     +10 science (capital) 

I   Tithe The Wealthy              +10 gold (capital)
T   Raise Monuments              +5 stability (per wonder on capital)

L   Bear Not False Witness     +5 stability (per territory on capital)

Tier 2 - Ascending out of the Ancient Era with enough followers (or when you have enough) allows you to choose.

C   Reject Luxury                     +1 industry (per follower; capital)

P   Seek Wisdom                     +1 science (per follower; capital)

L   Give Alms                           +1 money (per follower; capital)

 I   Observe Fasts                    +1 food (per follower; capital)
T   Mandate Patronage            +3% stability (per coreligionist state)
A   Scheduled Meditation         +1 stability (per follower; city center)

Tier 3 - Ascending out of the Classic Era with enough followers (or when you have enough) allows you to choose. 

I   Eschew Gluttony                    +5 money (territories)

L   Be in Harmony with Nature    +5 science (territories)

A   Steal Not                                +8 stability (territories)

C   Smite Unbelievers                  +3 influence (territories)
T   Theocratic Practices              +1 stability (infrastructure)

P   Moralistic Sympathies            +5 food (territories)

Tier 4 - Ascending out of the Medieval Era with enough followers (or when you have enough) allows you to choose on of the following.

W   Beware False Prophets    +100% turns before territories under your religion’s influence are converted by another ideology. 

H    Proselytize Daily    -50% turns before you can convert a territory that’s under a foreign religion’s ideology.

Tier 5 - Ascending out of the Early Modern Era

C -10% production cost on maker quarter
L -10% production cost on farmer quarter
I -10% production cost on market quarter
P -10% production cost on science quarter
T -10% production cost on common quarter
A -25% production cost on harbors quarter

Tier 6 - Ascending out of the Industrial Era

C   +5% per / co-philosophical - industry
L    +5% per / co-philosophical - food
I     +5% per / co-philosophical - money
P    +5% per / co-philosophical - science
T +10% stability per / co-philosophical
A +3% influence per / co-philosophical

This is still mildly abhorrent to me but at least now you've got systems layering on systems in an ever more complex simulator that should feel appropriate in an Amplitude game. And yes. I recognize that both suggestions basically remove the 'religion' and I think for now that's good. I think any faith system you introduce is hardly going to satisfying anyone and even here I'm still doing a LOT of fudging to approximate another trend in human history. Religious movements and the faithful would probably be difficult to reflect accurate in this game as a lot of those ideas are also cultural and on occasion ethnic in nature. You've made the decision to encompass large political and economic bodies and while I appreciate the effort to acknowledge the religions of the world I think it's a mistake to have it in the game. While religion has moved through time and through different cultures and brought it's own illumination to different cultures and countries over the years it's maybe not appropriate in a game which spans so much history with so many unique religions and views and cultural practices that aren't neatly simplified. Even the above system which I try to bridge philosophy to ideology and society seems intrinsically faulty. 

Although part of that might just be the groosly simplistic generalized and broad "merits" or "ethics" used that aren't historically interesting. Perhaps if this system was used you could set your philosophers to finding better examples from history of similar sentiments. I admit to being a little smug to sneak in a little Adam Smith with Moralistic Sympathies and +5 food (territories) because he was in his first principals a moral philosopher and his identification of the 'invisible hand' focused on a lot of practical day to day market behaviors that made sure people had food and shelter.

I've changed my mind about this system entirely. I thought I was contributing to the conversation by trying to shift away from problematic issues of state religion and cultures not being given the space to exist with their own religion but still connect your system to the built in game by recognizing that there are large cultural / philosophical values that people who identify as the same nation might internalize and pass down or pass on to their neighbors. But from the comments in discord apparently the idea of a state religion determining the values without question or challenge for their population is entirely normal and acceptable and spiritual autonomy is a myth. The Hittites were just as easily manipulated into believe and behaving one way as the Zhou by ... a vague bodiless body of priests who secretly guide magical belief energy into holysites for the glory of the Endless.

Remove the system.