I highly recommend Stan Collender's latest on GOP budget strategy for those who aren't quite sure what a "budget" (that is, a Congressional budget resolution) is and why it does or doesn't matter.
I think I disagree with him, however, on the question of whether "no budget no pay" was a mistake for the Republican leadership.
Not on the part having to do with the Senate; I agree that pushing the Senate to pass a budget resolution is basically a waste of the House's time. But on the other hand, I don't think it matters much. Sure, if the Senate actually passes a budget resolution this year then Republicans will have lost a talking point, but (1) given that the talking point was nonsense I'm not sure it matters; and (2) even if the Senate is careful to be as vague as possible -- and they might not -- it shouldn't be hard to develop new talking points out of whatever they do. I'd call that part of it a wash.
No, the real advantage of "no budget no pay" has nothing to do with the Senate. It has to do with the House, where the Republican leadership has to balance between a bunch of Members who want to pass the most extreme budget possible and a probably larger group of Members who want nothing to do with it. The leadership, which needs the crazies and crazy fellow travelers from time to time, wants to give them their budget resolution...and "no budget no pay" will help them pass it. Or at least, it certainly should help.
More generally: there are plenty of things that Congressional leaders want Members to do that Members don't particularly want to do. In normal times, budget resolutions are probably one of those; Members want it done, but there's not much of an upside in voting for it. Whether intended or not (and I have no idea whether this was thought through or not), "no budget no pay" gives House leaders an extra weapon for keeping Members in line for what should be a tough vote. In my view, it's a terrible precedent -- legislators shouldn't have a personal stake in specific votes (I called it flat-out corruption last time I wrote about it). And I have no idea how strong an incentive it will turn out to be. But either way, if it has any effect I'm pretty confident it will be over on the House side.