What's a US Trust then if it's not a body corporate incorporated
A body corporate implies a legal structure owned by its shareholders or, as the Yanks prefer, its stockholders. We have managed to invent bodies that are incorporated but not owned by share or stock holders but we don’t refer to these as corporations. For example in the UK LLPs.
A trust on the other hand is not a body corporate in the normal sense and nor is it incorporated in the normal sense. There is no public central register of trusts in the UK or for that matter in the States. We are familiar with trusts used extensively for estate planning and it is no different State side with Credit Shelter Trusts. Inevitably trusts in their various incarnations provide a useful tool for sheltering assets without necessarily incurring some of the disadvantages of a corporation - hence their wide spread use.
Different animals for different purposes.