I am going to try to answer your question without sounding abrasive or judgemental, as I understand where you're coming from, though without knowing your history with owning birds (Is this your first bird? First conure?) I'm not sure how much of this "issue" you're having is actually the bird acting out of character or just you not knowing how loud conures in general can be.
Conures in general are loud birds as a whole whether you're doing everything correctly or not, it's just how they are. That being said, green cheek Conures do tend to be the quietest conures, where as Gail stated Patagonian conures, Sun conures, and Jenday conures tend to be the loudest, no matter what you do. I've owned several birds throughout my life, but the only conures I've ever owned were both green cheeks. My first was a female cinnamon turquoise who passed away suddenly from a freak accident at about 9 months old. I now have a male, yellow-sided green cheek conure, and he is almost a year old now. He is definitely louder than my female was, and from day one it was evident that he was going to be a louder bird. I wasn't surprised about that as male birds do tend to be more vocal in general than female birds (there are always exceptions). I have heard horror stories about sun conures and Patagonian conures and the constant screaming, and as far as relating that to your male green cheek, because you said he was "not really screaming, but almost", that's a definite indication that your green cheek is not at the level of the typical sun conure or Patagonian conure because you would know 100% that he was screaming and would state as much, lol. There's not gray area with a screaming sun conure or Patagonian conure. So based on this info I'm going to guess that your green cheek isn't screaming and though I hate to say it, probably is not out of character in his noise level for a male green cheek.
I totally agree with your statement that the older your bird gets the more attached he is to you, that's typically how bonding with a pet bird goes, especially a green cheek conure, also known as a "Velcro Bird". They didn't get that nickname for nothing, it's very accurate, and the older he gets the more attached he will be to you, wanting to be with you 24/7 except when he's sleeping. That's how green cheeks are, and most people want this when they decide to get a green cheek. If someone asks me what type of bird they should get I first ask them what they are looking for in a pet bird. If they tell me they want a clingy, cuddly bird that will bond closely with them and be their best friend, the first species I recommend is a green cheek conure. They are definitely not independent birds, and are not at all happy being alone, no matter how many toys they have or what diet they're on. They want to be with their person all the time, and will let their person know when they aren't happy.
Just the same as the bird gets more attached to you the older they get, I've also found this to be true with their noises. My green cheek has absolutely gotten louder as he has aged. He hardly made a peep when I first got him at 12 weeks old, and now at a year he's constantly singing, talking to himself, talking to me, talking to my other birds, squawking, and yes, occasionally screaming, especially when I leave the room or first thing in the morning before I get up to greet him. It sounds like you're doing everything correctly, good diet, ample out of cage time every day, ample time with you every day, lots of toys and different toys often, so that's all good. And not wanting to encourage his noise is key, like you already stated you certainly don't want to uncover his cage while he's squawking for you, and you don't want to walk back into the room if he starts screaming when you leave. So you seem to understand how they think I'm trying to get what they want when they want it.
Have you had him to his avian vet recently for a well-birdy checkup? I only ask because I want to cover all the bases, and sometimes there is an underlying physical cause to elevated noise levels from birds.
You may also be dealing with hormones, this is also a common cause for them to be very, very vocal. Getting him on a SOLAR SCHEDULE may help your problem greatly if hormones are playing a part in the problem. Make sure his cage is near a window and that he can see the natural sunrise and sunset. Uncover him at sunrise, and put him to bed and cover him when the sun starts to set, which is much earlier this time of year, depending on where you live. He needs to get 12-14 hours of sleep every night to get his hormones in check, so if he gets up at 7 then he needs to be in his cage covered between 5-7 in the evening.
Short of trying those suggestions I hate to say you have a male conure and they make a lot of noise. I wish I could let you hear my guy right now as I type this. He's sitting 2 feet away from me playing on his cage top, and he's deafening my left ear...
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