One of the ways the president is limited is that the he can't authorize people to commit crimes. e.g. he couldn't instruct his AG to open an investigation into his political opponent under false pretenses in order to hurt his electoral chances. If the AG were to do that, it would be a crime. So the question isn't whether the president has authorized Musk to do something, but whether or not the president even has the power to do the thing he delegated.

And what is the power in question? It's control over spending appropriated by Congress. And that's where separation of powers comes in. Congress is supposed to control the purse strings, and the president is supposed to make sure the money is spent on the priorities of the people, taking care of prosecuting fraud and abuse. The point of giving Congress this power is to give the people a mechanism to set their priorities on how their own money is spent. It shouldn't be the case that one guy comes in and then gets to decide how to spend all our money.

But that appears to be what they are trying to do, in claiming that their cuts are all under the guise of reducing fraud and abuse. But really what they're trying to do is do an end-run around Congress. They want all the money, but they don't want to have Congress vote on it, because they don't actually have the votes to implement the agenda they want to, since Congress is so divided. So instead they're just taking the funding they have and allocating it in ways that support only the agenda items they want to see implemented.

What Trump and Musk are doing with DOGE and the federal payment system is in blatant violation of this separation of powers.