The House holds the power of the purse, but the President is the sole holder of Executive powers. That is to say, Congress legislates the budget but the actual execution of that budget is up to the President.
If Congress feels the President isn't executing the budget as legislated, Congress can impeach the President. As for the judiciary, SCOTUS already ruled that the President enjoys absolute immunity in the course of executing his Constitutional duties.
So far the House "applauds"[1] DOGE, so all is well.