One thing I think people forget about the President is that he is not, constitutionally, a leader. He is an executive.
People get wrapped around the veto, and the fact that with the staff, and the executive ownership of the departments, the President proposes the budget.
But the President also administers the budget...
A few examples,
Say the President thinks we don't have enough meat inspectors. The military does what the president says. The congress chooses the funding of the military, but not their activity, other than the power of approving wars. If the President sent the Army Veterinary corps (who inspect meat for military consumption) to meat packing plants, he would be within the law to do so.
Another example is timing. You see, Congress says what amount of money will be spent, and on what, but not when. Imagine that the President says "We will cancel all military contracts today (using the clause in all military contracts that lets the US gov't cancel it at will), and re-open them when the debt ceiling is raised above the estimated spending for the year."
The president can tell the military to not spend one dime more than they need to for paying the troops, to cancel all repairs in shipyards, all purchases of equipment and vehicles, etc. This would be a major reduction in spending, which would buy the government weeks in which to resolve the crisis.
Republicans seem to think only the military deserves funding, well, the military is capable of doing a lot of jobs besides defense. China maintains their huge army by having them do non-defense work. Maybe we should use the military to cover every cut the republicans make in the non-defense budget?