The “Department of Government Efficiency” isn’t a real government office and as such has no actual power. Musk and Ramaswamy won’t be able to do anything, and the budget is controlled by congress, not the executive.
The key thing here is the "Due Care Clause." The administration is required to implement our laws with "due care." If there's a law on the books the administration is required to enforce it. The means and expected effectiveness of that enforcement are often not specified.
I believe you're referencing "budget resolutions" which are not law and have a more complicated relationship to the process.
... who promptly pronounces it "dead on arrival".
Which isn't actually true. They still use the executive branch proposal as a baseline. They, too, are supposed to authorize payment for the programs that they've passed laws to create. Usually, there's somebody in the room whose constituents benefit, and they can insist that it continue to be funded.
And of course this time around, the President will have a Congress who is controlled by his own party, so the relationship will be less hostile. In all likelihood they will end up putting back some programs instead.