Questioning the Power of the Line Item Veto

Dear Representative Platts;

I'm trying to understand the legislation concerning the Presidential Line Item Veto. So here's what I understand:

Let's say the HHS has requested 1B for Heating Assistance for the Poor during the budget/appropriation process.

If we grant the President a line item veto he could eliminate this spending.

By eliminating this spending we would reduce the federal deficit by 1B.

The 1B would no longer be available in the total discretionary budget for any program in any department.

Am I correct up to this point?

Can congress override the veto?

If congress can override the veto, why wouldn't they? They have already agreed to the spending prior to sending it to the President the first time.

If congress can't override the veto, that would seem like a serious imbalance of power.

As an alternative to the "Line Item Veto", which if I understand correctly is a method of attempting to control "pork-barreling", why not just restrict all legislation to the primary purpose of the legislation instead of allowing ammendment after ammendment of unrelated spending?