But sentences starting "People believe ..." are common in grammar books.
People believe all kinds of things. But "People believe thousands of birds died" is unnatural, because we can't be talking about people in general. It means some specific people, and you need at least "The people believe ..." or "Some people believe ..."
You have to change such a sentence into personal construction and the right answer is "He is believed to win" or "The missing jewellery is believed to have been found"
As a learner I have no idea what sounds unnatural and what doesn't.
But from a purely grammatical point of view, "He is believed to win" or "He is hoped to win" is correct, isn't it?
It probably sounds silly, nobody uses such sentences but theoretically it's ok. Right?
"He is believed to win" is wrong, as is "He is hoped to win." Grammatically wrong.