When the Republicans controlled the House of Representatives, they were criticized by many political scientists including Barbara Sinclair, Thomas Mann, and Norman Ornstein for attempting to completely shut the Democratic minority out of the legislative process. While many Democrats (including the President) have expressed a desire to be more engaging with the minority party, Politico reports that Democratic leaders are planning to strip the minority party of an obscure, yet important prerogative: the right to offer a motion to recommit legislation to its originating committee with instructions to amend the legislation and report it immediately back to the House for final consideration.
Although it rarely gets the attention that the sexier legislative procedures like the filibuster, the hold, or Calendar Wednesday, the motion to recommit (MTR) is provides an important opportunity for the minority party to participate meaningfully in the legislative process. Essentially, when a bill is passed, any opponent may make a motion to send the bill back to committee with some suggested amendments. But because the language of the motion may require the committee "to report forthwith," the MTR is essentially an opportunity for the minority to offer substantive amendments (but ones that may be again amended by the majority). So even if the Rule Committee doesn't allow it to offer any formal amendments to the legislation, the minority will still have an opportunity to at least force a vote on one of its amendments, presumably one with the greatest importance and/or chance of peeling off enough moderate majority-party voters for passage. Even though few MTRs pass, the threat that they might pass or at least force some unpopular votes generates some leverage for the views of the minority party. As the Politico article points out, it was the threat of an embarrassing vote on an MTR that pressured the Democratic majority to vote to suspend the moratorium on offshore oil drilling.
It is also important to note that the MTR does not just protect the minority party, but it strengthens moderates of the majority party by giving them a credible threat to vote with the minority on the MTR if their concerns are not addressed in the bill. So the Democratic leadership's attempt to eliminate the MTR is also an attempt to shift the balance of power in a progressive direction. Moderates should undoubtedly oppose the rule change if they think bipartisan governance has a chance to work, but, equally certainly, they will be under intense pressure from Democratic leaders to support it.
I think further efforts to foreclose minority participation are likely to be counterproductive. The House of Representatives suffered greatly as an institution due to the heavy-handedness of Republican leadership, and it would be a shame if the new boss was the same as the old boss.
P.S. I should note that there is some academic controversy as to just how much leverage the MTR gives to the minority party. Those who are interested should see this article by Keith Krehbiel and Adam Meirowitz and this one by Jason Roberts.