Aww he just wants to be with you but I get it, you can't be there 24/7. Is your cat elderly? Our cat increased in her meowing volume and clinginess as she got older. She lost hearing and couldn't tell how loud she was! She wanted to be with us because she couldn't walk without discomfort. Wanted to be carried around, she didn't like the cold slippery wood floor. She was 17 when this behavior started. Her joints were sore so we gave her cosequin. She called for us less when she was feeling better. Then she was going blind and she became anxious. We gave her CBD oil treats, and that helped for a while. She was more relaxed and happy. but a few years later she eventually went totally blind from age and would poop in the wrong places. She found a place to hide and passed away at 20 years old. Even though we had her for a long time, over half of her life, we think back on her and wish we could have had more time with her.
I know you said don't recommend playing or cuddling with him, but he really does want to be with you. I think the main thing is that your cat is lonely. It really would help to give him attention for at least 15 minutes a day of intense play, right before feeding, then go outside. his clinginess could have increased especially if your cat has recently lost a companion, is very young and is used to you or momma taking care of him when he was a kitten, or is very old and is experiencing dimentia or has an underlying health issue he is trying to tell you about. Check him over, see if he might seem sick, has an injury you can't see. consider if hes exhibiting other behavior issues like urinating where he shouldn't or not burying his poop.
Can you perhaps harness him and bring him with you when you go out with your birds? If not and he has to stay inside, try to put on some cat tv (there are great videos on youtube your kitty might enjoy), or see if giving your cat a wall mounted toy on an elastic string would occupy him. Try some nip for his entertainment, or treats with cbd oil for a calming effect. Offer some barley sprouts or even give him a cardboard box to play with. A change in environment, something new to explore. We wish we could be there all the time but have to be realistic, and it's our job to offer our furry friends things to do when we can't entertain them haha