In practice, virtually all groups and organizations and board and committees do have motions to close debate. A chair can suggest that closing debate is a good idea, but cannot force it as you describe. The remedy in parliamentary motions to this situation are motions such as point of order, appeal from the decision of the chair, adjournment, etc. Also, you could bring the problem up at a meeting of the plenary board which created the board. What authority a board has, as well as its entire nature, is described in the bylaws which create it.
Boards vary considerably from organization to organization and circumstance to circumstance.
It is difficult to make generalizations that apply reasonably to them all. Chairs use a variety of means to facilitate the fair allocation of the floor. Organizations sometimes adopt standing rules or special procedures to facilitate speaking, especially in large organizations. There is no requirement that a motion have anyone speak on it, either for or against. The requirement is only that the chair provide opportunity for debate.