The Founders didn't anticipate riders being inserted into omnibus bills. The dissenting opinion in Clinton vs New York stated that he supported the line item veto because each specific appropriation could be considered a separate bill - an omnibus bill with 1000 appropriations is effectively the same as 1000 bills with one appropriation each. The omnibus bill is designed so that the President doesn't have to sign 1000 bills - he/she just signs one bill.The "Line Item Veto Act", was signed into law by Clinton in 1996(he used it 82 times). It originally gave the President the power to remove/veto any part of a bill passed by congress(pork barreling, line items, etc.). And, to decrease federal expenditures and combat wasteful spending. Without having to veto the entire bill. In 1998, the Supreme Court struct down the Line Item Veto Act, as being unconstitutional. So, unless you want to add a new amendment to the Constitution, a "Line Item Veto", is a dead subject. The constitution had intended the President to have the power only to veto the entire bill, NOT simply part of the bill. Otherwise, IT WOULD BE THE PRESIDENT LEGISLATING THE BILL, AND NOT CONGRESS. The President can already veto a bill, and then suggest to congress the changes he wants.
The problem with the current system is that each bill isn't restricted to a specific topic. Omnibus appropriations bills contain riders and other unrelated matters, which means that the President is obliged to accept whatever is in the bill otherwise risk shutting down the government.