No, Article 1, Section 8, Clause 1 has nothing to do with the budget. It has to do with paying debts and collecting taxes.
Article 1 Section 9 Clause 7 is closer. It says Congress has to pass an appropriation before spending money. That’s still not budgeting.
The 1921 Budget and Accounting Act says that the President has to submit a budget proposal. Congress modifies and passes it. The President signs that or there’s a continuing resolution.
No, Congress doesn’t control the budget, either Constitutionally or procedurally. The President proposes one and generally gets most of what they want.
In fact, Congress doesn’t even pass a budget. The President proposes a budget, required by law, and Congress passes appropriations bills, required by the Constitution.