Do you always need probate court?

You mention a trust. One of the purposes of a trust is to avoid probate. However, the successor trustee should see an experienced lawyer who can guide the trustee through the administration.

Yes, by law when intestate or will.

You have confused terms so I am not sure what the situation is. The Trustee of a Trust divides among the Beneficiaries, according to the Trust instructions, the assets of the Trust; there is no court involvement unless someone disputes the terms of the Trust or what assets are in it. An administrator of a Will, who might be the executor named in the Will, but who must be appointed by the probate court judge, carries out the terms of the Will as to the estate left [all items not part of a valid Trust]. Technically, all Wills must be probated, but if there are no creditors, no potential heirs who disagree with the distribution, and no titles to vehicles or real estate that must be changed, then there is no one who would complain about the lack of probate.