Discharge Petition

There actually IS one way that a bill can be brought for a vote even if it does not get reported out of committee and is not selected by Leadership: a discharge petition.

If at least half of the Members of the House (218) sign onto the petition to discharge a bill, it is brought to the Floor for a vote.

Remarks of Pope Francis to a Joint Meeting of Congress, September 24, 2015

You are the face of its people, their representatives. You are called to defend and preserve the dignity of your fellow citizens in the tireless and demanding pursuit of the common good, for this is the chief aim of all politics. A political society endures when it seeks, as a vocation, to satisfy common needs by stimulating the growth of all its members, especially those in situations of greater vulnerability or risk. Legislative activity is always based on care for the people. To this you have been invited, called and convened by those who elected you.