L.C.
Wow... dedicated little guy! Of course, I knew one little girl very like this: quite capable of screaming for 10 hours straight. Goodness knows why anyone felt the need to provoke that discovery...
Some people are placid. If they don't get their needs met, or never feel heard and understood, or have something wrong for long enough, they just figure 'oh well, this is never going to change, no point in making noise.' I am sometimes much more frightened for the young when they are placid than when they feel free enough to make a lot of noise.
Other people are not placid and are convinced that the only problem is that they haven't said the right thing yet...so they keep trying and trying and trying to make it clear what it is that they need. They will try, literally, for hours and hours and hours. As you have experienced.
The answer is fairly simple, but it feels like 'giving in' or 'letting the child win', both of which come from problematic mindsets.
What exactly is wrong with a child getting to win? When that 'win' means that the child gets what the child NEEDS. Sure, to an adult who knows how to wait and why everyone else on the planet is so much more worthy and important than themselves... it feels wrong to just let a child have what he needs. But making another person suffer because we always have isn't a very loving thing to do. It does nothing to alleviate our lifetime of suffering, either. No one 'loses' when a child 'wins' in getting his need met.
There is a fear that if a child 'gets what he wants' all the time, he'll be insatiable. But really, it's the other way around. It is not people who have ready access to everything they need who become the greedy gluttons of the world: it's the people who've had all the 'good stuff' held away from them, used as bribes, and removed as punishment... those are the people who are grasping and selfish and constantly focused on making sure they get 'their fair share.'
I might suggest that multi-child care is inappropriate for this needy little guy, as you rightly say you simply do not have time to deal with him all day, every day. Be assured: his parents didn't 'make him' this way any more than you have. He is who he is, even if that is someone who is hard to love.
Like everyone else, he needs generosity and love and affection and to get what he needs...even if he seems to need more of it than anyone else thinks is rational. He's not rational --he's a toddler. And he needs what he needs, when he needs it. Rational doesn't enter into it.
Some of the most secure, independent young adults I have ever known were the ones who were so seemingly insecure, so noisy about getting their needs met as young children. The very same children whose parents were regularly criticized for 'giving in' and 'letting them rule' by giving them what they needed. The 'spoiled' children are, in fact, the most mature, most generous, most balanced and stable I have ever known.