Want a more engaged Board? Embrace feedback. Don't ask the Board to rubber stamp decisions that have already effectively been made by Board leadership.
The collective wisdom of a Board is the most valuable asset a nonprofit organization has in its Board. Don't waste it.
Before putting a Board matter to a vote, ensure plenty of time for adequate public and private feedback. Bring a Board matter up at one meeting (or other communication), vote on it at another. Many people will not speak publicly during a meeting - they feel more comfortable following-up privately.
Don't risk losing valuable feedback on important matters due to a rushed, uncomfortable environment.
Board leaders will take these types of tactics to control the ultimate result and avoid conflict. Maybe there are too many people in the room - too many tangents - too much chaos. Don't let these realities deter you as a leader from getting the feedback.
While you can't please everyone, you just may wind up tweaking your Board actions based upon feedback for the better and in the end your Board's collective actions will gain more support from the Board members, resulting in a more engaged Board.
Board members will burn rubber to avoid Board meetings when their input is not valued, deeming meetings a waste of their precious time. Just being asked to vote yay or nay or a matter, is not considered by most to be valuable input.