I find with my sunny that if I am available when she has access to toys, I am preferred lol. She will happily swing and nibble if we are in the same room though and she loves to go to her play gym and swing around in there....then her natural branch and tootle about on there with the toys hanging on that....
Cal favours toys she can climb all over as well as chew....she loves to swing on long ropes with wooden blocks.
She adores her toys, but I think when she is alone she probably doesn't bother too much. She is usually sleeping when you check on her haha! It's different here though, as we both work and live in the same building so she is always with us.
As for the screaming, I combatted that by teaching her no shout using a combination of phrases, praise and whispering (this works as they are nosy lol). Now our routine is this:
Be a minute! (I exit the room)
Instead of screaming Cal sounds like a smoke alarm running out of batteries....beep...beeep....beeep.....
I come back in: Good girl, wasn't long

Cal: Beep beep beeeep. Hello!
She sometimes decides shouting is in order in a random burst occasionally, she does a short blast when she wakes up and a short one while you sort out her breakfast and when she's alerting you of something!!! (Very handy for when I'm in my stock cupboard in the shop and a customer comes in

)
Calling to her flock is very important and she is very intune about what everyone is doing. She beeps if she hears the dogs bark, says hello when the cat enters the room, says hello to the rabbit and of course greets us enthusiastically. If the toilet flushes....she beeps....if a door is opened...she beeps....she needs to know where we all are and that is what I kept in mind while teaching her no shout. If she beeps - we respond.
She will beep periodically until the person who has left the room has made their intentions clear. If I go to the loo, she beeps until I appear back. If my husband leaves to go out, she beeps until she hears the back door has been shut and she knows he will be gone for longer than a few mins.
I think it would be very reassuring to your Iago if you could recognise this and form your own trigger word to soothe that you will be back, you're only in the next room or w/e and you're "reporting" back

hehe
I hope I have been some help as I understand how difficult it can be to get inside a sunny's mind!