I like the approach taken here, however have 2 main changes that need to be made (before I make comments on specific formatting changes).

1. The error capture decorators are used for any interactions that do not have exceptions captured by default (I.e. tasks & event listeners need exceptions capturing, whereas commands have exceptions captured by the on_application_command_error event listener). Therefore handling of capturing the MessageSendForbiddenError should be performed by the decorators for the tasks & event listeners as it is now, but also is required within the command error handler.
   TL;DR: handle MessageSendForbiddenError within on_application_command_error event listener.

2. The current error capture decorators use the capture_error_and_close() function. This should be refactored to determine whether TeX-Bot actually needs closing on a per-decorator basis. The capture_guild_does_not_exist_error & capture_strike_tracking_error decorators should close TeX-Bot, whereas capture_message_send_forbidden_error decorator shouldn't.