Why doesn't funding for U.S. government services simply continue at its previous level if a new budget can't be passed?

Question

In the United States, a failure to pass a budget for the new fiscal year results in a government shutdown, during which all government agencies lose funding except those which are funded through other avenues (for instance, the Postal Service, which funds itself through the sale of stamps), and all employees of now-non-funded government agencies (except for those very few whose work is considered essential, such as air traffic controllers) are temporarily laid off.

Why not make it so that, if a budget can't be passed by the time the new fiscal year starts, funding simply continues at the same levels as the previous year until such time as a new budget can be passed?

Answer 1

Some parts of the government do indeed work that way. Most notably, Congressional salaries

Unlike other federal employees, senators and House members continue to get paid through a government shutdown. That’s because their salaries are literally written into the Constitution. Article 1, Section 6 specifies, “The Senators and Representatives shall receive a Compensation for their Services, to be ascertained by Law, and paid out of the Treasury of the United States.”

But the reason why level funding in the absence of a budget or appropriation can't happen is the Antideficiency Act, which explicitly forbids spending that Congress has not provided funding for

The Antideficiency Act (ADA) is legislation enacted by the United States Congress to prevent the incurring of obligations or the making of expenditures (outlays) in excess of amounts available in appropriations or funds. The ADA prohibits the federal government of the United States from entering into a contract that is not "fully funded" because doing so would obligate the government in the absence of an appropriation adequate to the needs of the contract.

Answer 2

Congress will occasionally do "continuing resolutions" (CR) which provide funding based on previous year funding.

It would seem that they should be able to pass a law that stipulates an "automatic CR" such that if a consensus cannot be achieved, then a CR is automatically passed. This would solve the problem.

But, then they wouldn't get so much media coverage. And that is the primary issue. Politicians love shutdowns because it gives them so much free press.