Only people assigned to this task can submit time

When trying to add a task to your timesheet, the task is grayed out. Hovering over the task displays the message - Only people assigned to this task can submit time

When trying to submit time to a task, it is instantly rejected with the message - Sorry, only people assigned to this task can submit time. Please contact your project manager.

These messages are shown when the project flag Resources may report time only to assigned tasks is enabled. To fix:

Disable that flag and refresh timesheet
Assign the resource to the task from the Project Task Planning Tab