I had Consulates but I'm pretty sure that CS friends only give food to your Cap. I'm lazy and stupid and never bothered to check. The happiness is virtual growth for your satellite cities though since it means you can grow bigger faster usually.
Are you rolling "we love the king days" in your cities are frequently as humanly possible? I 100% never sent a single food caravan to Te-Moak in any of my games. The only city that I ever send one to is my Cap and I've basically never sent more than 1. All I did at Te-Moak was work my highest food tiles (3x Salt 2x Wheat) and then worked on a whack of farms. That + Granary + we love the king + eventual Fertilizer was good enough for 23 by 217 and that's taking into consideration that I was working a ton of specialist slots. I work every science, artists and engineer slot in my cities as soon as I can basically.
I 100% didn't have a religion that game. I had the +10% growth pantheon but that was loooooooooooong gone as soon as the first religions started coming out. I think that the only thing that you're missing is a constant stream of "we love the king days" because I did absolutely nothing special, had no religion, didn't have HG/Artemis and I mean that city had no access to fresh water but it still got to a decent size.
It's tedious but you basically just have to prop the diplo tab open to scan across the civs and CSes and find manageable ways to get luxuries that you need. I'm not going to lie and say that it's easy, it's a pain, but you need to put up with it if you want fast times. It's going to cost you a lot of gold and happiness but I mean that's life. You need to do your best to make CS allies to create happiness buffers so that you can trade your luxuries and gold away more or less. Takes a bit of practice but I mean the results are obvious.
For what it's worth all of my fastest wins involve games where I get Oracle and subsequently Consulates which makes dealing with CSes easier. Oracle is fairly easy to get, even on Deity, especially if you can work a mine or two. Getting it pre-NC and still getting your NC by turn 100 is do-able on some starts (not all obviously) and that's what I typically shoot for me if I want to have a sub 240 win. The only terrible starts are Jungle but those are always bad and you'll just plain never have amazing times in jungles I don't think. I could be wrong but I mean I've never had a "good" jungle game.