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. Voting Dear John, I need some advice regarding parlimentary procedure. I am an elected member of a six member school board in Franklin, Tennessee. It is customary that the chairman votes on all matters since all members are elected at large and not by districts. In a board or committee, the chair does vote, can debate, and make motions.