Congress is supposed to fund the government and say how money is spent.
The executive "cutting" things to "save money" is basically the executive assuming the power of the legislative branch.
That said, what we're witnessing doesn't actually seem to be about spending. It seems to be more about obtaining direct dominance over the whole branch so it can be run essentially free of any oversight or connection to legislation and law - IE to create a quasi-imperial executive branch narrowly focused on the priorities of it princely leaders. So most of the "firings" are basically to make a point, winnow things down to loyalists who will ignore the law, and keep the news so filled with surface reporting on each new small outrage that the big ones don't get noticed, not so save any money.
Anyone who truly believes that there's tons to cut and that government institutions need reform can't also think that getting rid of huge swaths of the institution without attempt to identify improvements, priorities, or waste, will actually create efficacy, unless their real goal is not reform and cutting waste, but rather to make the whole of the current form of government fail.
I've seen some wild social media posts that suggest that people have really absurd views on this, too - someone I went to high school with like 20 years ago is having some kind of issue with navigating the Social Security Administration for something via phone and he's cheering on what's going on like it will actually solve his issue.