Well let's see, a few thoughts from looking at things.
First, you're not wrong by wanting to play tall or do things the tall way. More pop is always good
It looks like you have 9 settled cities. For turn 134, I'd classify this as more of a wide strategy, not necessarily ICS. Which is good I think, ICS is a lot harder to pull off if you're not Maya. ICS I'd say is roughly one city per turn 10 turns. But with the happiness on this map, i think you'd have been better off with 1-2 cities fewer, at least for this stage of the game.
In your settled cities, you have fairly few unique luxes and few luxes in general. That probably slowed down your start considerably until you got some of these puppets with luxes. A taller strategy might have suited this map better.
You have 6 workers for 13 cities, which is just not enough. Especially when you have 2 in a size 1 city (1 will keep up with growth) and 2 up by a puppet. Meanwhile you have a city like Cumae which has only 2 improved tiles and is size 5. That's a ton of wasted food or hammers. You're also have some poor tile choices - for example you have a riverside marsh locked in at 1 food 1 gold when there is a 4 food 1 gold tile available or could purchase a hill, or best yet, lock that guy in as a scientist. If you want to stop growth, work a farmed river grassland and a mine or scientist in pairs. 4 food feeds 2 citizens, use that 2nd citizen for optimal hammers or science growth. When you're not stopping growth, lock in a lot more of those 4 food tiles! I think I also saw an unemployed citizen. Just overall, you're working too many unimproved tiles, throwing away tons of hammers and food.
It's great that you have 3 universities up, but you're not working all the specialists. Part of that is the low citizen count, part of that is the tile management and lack of workers. If you had 10 more mines up across your empire, you could be getting all those structures up all the quicker.
It looks like your cities are a little on the smallish side. OK for an ICS type strategy but for wide you still need some cities to grow. Even in a wide strat it's good to try to have your capital be close to 1 pop per turn 10 turns. And by this time I think some of your secondary core cities should be size 6-9, hopefully not 5. I suspect this is most related to the lack of workers, but also the happiness issues as well. Also you can't neglect granaries and aqueducts just because you're not tradition. 80% of the time, granary is the first building built or preferably bought in a new city for me, except if I'm playing Maya or Ethiopia. You mentioned you are limiting your growth to control happiness, which I don't think is the best idea as I mentioned above. But if you really want to pursue ICS, you need to get universities up in all the size 5 cities ASAP with at least 1 scientist running. If you stop expanding and grow a little taller, you can delay them a liiiittle longer until you grow a little bigger and get some excess happiness.
Messenger of the Gods is OK, but it looks like +2 faith from quarries would not have been bad, with 5 quarries in your first 3 cities and at least one other. I do see another civ has it, so it's possible it was chosen, but it looks like this civ was late to the party with the beliefs they chose, so if it was available, it should have been considered. The science from MotG kicks in a little late, after the small amount would really be useful. Not a huge fan except on Maya with true ICS. Your religion is clearly struggling (and you might even struggle to remove it from your cities and replace it with your own and have it be stable); that extra faith would have meant an earlier enhance and faster missionaries to spread it around. You got itinerant preachers but your religion was still squashed. At least, it probably would have been better to get a missionary before enhancing. Increased range is really doing you no good at this point, and you're only getting 2 happy faces from temple in 2 cities. I also think the Pagoda pick is a little suspect without a faith generating pantheon. With how much spreading / eliminating you're going to need to do to get your religion off the ground, I don't see you getting many Pagodas built. I think you could have gotten a lot more mileage out of a religion like Stone Circles / Church Property / Religious Center / Guruship / Itinerant Preachers (or if you anticipated Arabia's religion taking off so much, Holy Order).
Hmmm those are the major things that I think could be improved on. I think the most important point is, you need more workers and to prioritize granaries in my opinion. I think your capital should be at least size 12, and 2nd-4th cities around size 6-9 at this point. This would make it much easier to work both university specialists in all those cities while staying productive and growing.
