Agreed. At least on the procedure. My point is about keeping it in public view as long as possable. Most of the public ignore this as a slight noise that is not wort their time to worry about. It needs to be elevated to their attention as was the resolution to cut and run. When it was a political game based on bluff the dems made hay. When the Republicans brought it to a vote it suddenly had everyones attention.
And even if you bring something to a vote in the Senate, the minority always has the opportunity to amend and muddy up the message, which is not the case in the House. Which is why you won't see a cut-and-run resolution offered in the Senate.