Yes, many forces, including the role of president, shape the federal budgets. But eventually and fundamentally it's the power of congress that control spending and overspending (deficts/debts). But congress is representatives of our voters. So, it's primarily our responsibility for having so much debts, not any president.
Read the Constitution, Article I, section 7, clause 1:
"All Bills for raising Revenue shall originate in the House of Representatives; but the Senate may propose or concur with amendments as on other Bills.”
and its Article I, section 9, clause 7 (Appropriations Clause):
"No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of the Receipts and Expenditures of all public Money shall be published from time to time.”