I’d say that having three different words for “because” increases nuance. As the link to merriam-webster’s article pointed out, you get a nuance of formality between “because” and “as”; “as” is somewhat more formal. I’m not sure if there’s another nuance between “because” and causal “since” but smart money is on there being one (if you survey the use of the two I bet you will find there are very subtle differences of usage there – there almost always are nuances of difference between supposedly synonymous words, even if they’re only differences like level of formality).
The way the text here it phrased, it sounds like McCarthy wronged the Democrats by giving them nothing, and they’re getting him back by letting him go. But the article makes it clear that they asked for nothing and he expects nothing from them, and respects that.
No need to imply animosity and back biting between McCarthy and the Dems that isn’t there.
It’s more like “you do your thing, we’ll do our thing, and that’s OK, no matter how it turns out.”