Agencies are responsible for the content of their websites. If the agency is shut down, there is legally-significant content online that is under no chain of accountability. Every web page on a government site is a potential liability. In the normal course of business (e.g. when the government is not shut down), every piece of content goes through some sort of approval process, or at least someone can be held accountable for it.
Imagine a law is amended, but FCC.gov still has guidance published on the pre-amended statute. Kind of a crude example, but I think it highlights the problems of not going whole-hog on shutting down these websites.
Moreover, it's pretty difficult to not defer to an agency's judgment when they have been winnowed from ~1700 employees to something around 12.