House is holding the purse strings
The House of Representatives holds the purse strings. It has ever been thus. That is the way our system was designed and it has worked for well more than 200 years.
Every year two bills are required to spend funds on any line item. The item must be supported in the Authorization Bill and the Appropriation Bill. Omission from either bill will deny the item. The House is doing what it must accomplish in the proper and lawful manner. There is no way for the Senate to override what the House has determined. The Senate may request a change to the budget - if they can get their act together in time. Their track record on passing proposed budgets is not what one would term as "timely." The House does not even have to consider their input.
All programs except "Affordable Care" are currently proposed by the House for funding. Any government shutdown (which is usually announced by the union shop stewards; I have heard them pronounce, "We're shutting down the government") will be completely at the hands of the administration, and on Democrat heads forever.