He has a point, Congress sent him the spending bills. The president doesn’t have the power to raise spending, Congress does.
The responsibility lies jointly with both the President and the Congress.
Personally, I favor a balanced budget amendment (initiated by the States) that would in essence forbid any spending above available revenues, excepting national emergencies and in those situations a 75% affirmative vote from both bodies of Congress would be required.
Of course, this would not keep Congress from attempting to raise taxes/revenue in order to justify additional spending, but that discussion would have to take place in full view of the electorate, as opposed to the current tactic of Continuing Resolutions and letting the national continue to rise unchecked.
The power of the Veto is on him.