"Should" or "are required to?"

Regulations are minimal compared to what most consumers would want and expect.

https://www.ams.usda.gov/rules-regulations/cool

For some things like meat, eggs, and milk: The USDA-inspected processing facility is required to be listed on the package. It is usually not very far-removed from the farm. It's coded, and these days it's usually printed near the sell-by date.

They (the USDA) publish cross-references betwixt the codes and the processing plants.

It's been kind of a pain in the ass when I've looked this stuff up in the past and I wish someone would take it upon themselves to make it easier, but it's do-able.

Depends on the thing. Olive oil, for example, is required to list its country/countries of origin