Reducing or withholding funding is a result of the separation of powers. The legislative branch is the source of funding and withholding funding for various programs has been done many times in the past as a negotiating tactic (e.g., which results in the Washington Monument strategy).
The Washington Monument syndrome, also known as the Mount Rushmore Syndrome, or the firemen first principle, is a term…en.m.wikipedia.org
In 2013, White House responded to the short-term funding loss by closing access to popular government sites (e.g., federal national parks).
The vast majority of the government spending and function is unaffected by the shutdown theater. I’ve read that at most 20 percent of the government is affected