No. It’s does make sense, as cases of attacks outside of business hours are harder to deal with because employees are off work. This is further supported by the fact that cybercriminals target websites most especially during off hours and holidays as it buys them more time before they’re discovered. Turning off the servers outside of business hours would effectively prevent this attack vector, at the cost of profit margins.

Most businesses understand that this is a vulnerable period of time and is why most of them mandate that their IT & cybersecurity professionals must stay on call even if they’re taking time off.

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I’m unable to find a source that supports the claim that the Social Security Administration states that “websites need to shut down outside of business hours”, please provide one.

The operational hours for the CBSV application and web services provided by the Social Security Administration are clearly outlined, indicating specific times when services are operational, including weekends and evenings. While software maintenance may occur on weekends, there is no indication that they encourage or mandate websites to shutdown outside of business hours as a standard practice; source.
Additionally, the Social Security Administration emphasizes that many services are available online and over the phone, even if local offices and the federal site are closed, highlighting the accessibility of online & offline services for individuals needing assistance; source.
The message on the “my Social Security” update page confirms that all services are currently available, with any planned maintenance activities or temporary issues being transparently communicated to users.

Therefore, while they have their own operational hours, there are a plethora of other services outside of the federally owned website including websites for local offices* and there’s no indication that the Social Security Administration mandates or advises other websites controlled by local offices to shut down outside of business hours, let alone any other branch of the US government.
In conclusion, you’re hyper-generalizing based on the practice of a single cherry picked department, in addition you have no knowledge on the actual reason they have operational hours in place to begin with and are speculating on pure assumption that it’s for security purposes when it could just be a matter of ensuring control without having to go through the usual red tape.

*(Local Social Security Offices don’t typically have their own separate websites but they can, the LSSO I use just so happens to be one of the few that do as they service a very large amount of people at any given time and are constantly short on staff.)