And if the president's party blocks action on the bill via one method or the other and it automatically becomes law?
I really do mean no offense, but this is a terrible idea. Presidents would be using it to screw with the opposition party come election time like you wouldn't believe. I would rather the minority party in Congress be given more power to force votes and more rules that force Congress to act in certain circumstances to prevent things like year-long SCOTUS nomination blockades or to prevent power from becoming too centralized (like it is now)
What you're really asking for is more centralization. That's what would make those things happen.
I've been considering if a parliamentary system is better. This seems like a good compromise.
And the President's party could only block a veto on the bill if they had a simple majority who voted to NOT veto it.