So, it turns out some Democrats are pulling for a procedure change under Pelosi, saying bills should be easier to see a vote when 3/5ths in the House are co-sponsors – it's more than a majority, so it stands to reason the bill would pass if it isn't completely sliced and diced in amendments.
They have some other demands, which you can read about WAY DOWN in this article. I don't know why the Chicago Tribune thinks what a policy is barely rates a mention in their inverted pyramid, but I do know I wouldn't make the same editorial choice myself. And the policies these Democrats want are super reasonable.
Some people have been referring to these are "pro-GOP" rules. They, emphatically, are not. They are rules that give individual Democrats dramatically more control over policy than they had before, compared to leadership on either side. That is the reason this debate is happening now. Because why hand someone buckets of power by making them leader, if that means you'll never get the things you want most, even when the American people and three fifths of the House agree with you.
I noticed a lot of prominent Democrats oppose it, too. I could not, for the life of me, tell you why the guys at Pod Save America oppose it, despite listening to them explain their opposition. I guess they think it'd empower the minority party, and now that Democrats control one chamber in Congress and neither other branch of government, they've decided that's outrageous, of course minority parties shouldn't be given any power at all.
Again, look at the rule changes. This gives power to people who have extremely popular legislation that party leadership opposes. If someone could do some anthropology on this for me, how non-legislative Democrats came to oppose this, I'm all ears. I'd love to hear the strongest possible argument. "It'd be chaos!" sucks as an argument, and that's the best one I can imagine.
My complete guess is that Pod Save America is against these rules for the same reason some of the newer Democrats are – because bipartisanship makes it harder to paint Republicans as lunatics. A careful read of their comments, as well as e.g. Ocasio-Cortez's statement in opposition make this pretty clear. So it's a political move pretending to be a policy move to oppose was a procedural move in the House pretending to be a policy move in order to appear less like a political move. That's my final answer, I guess.