I've thought for some time that a Commonwealth-style question time would be an excellent addition to American politics, but the President is not the one who should be taking the questions. In the House of Commons, the MPs are questioning one of their own, the Prime Minister, who is a member of the legislature, not the executive. The best parallel would be a question time for the Speaker of the House or Majority Leader, who are the #1 and #2 members of the legislature under the American system.