Moving Scheduler Module's 'Scheduling Options' Out of the Vertical Tabs in D7

...or, "Always Check Your Module Weights when form_alter'ing"

Scheduling options from Scheduler module

I spent about half an hour today trying to use hook_form_alter() to move the 'Scheduling options' fieldset (provided by the Scheduler module) out of my node form's vertical tabs (down where URL path settings, comment settings, etc. are jumbled together).

I couldn't even see the 'scheduler_settings' form settings when I looked at the form's array, even though I knew it existed (since it was being displayed, and the scheduler.module defined it using its own hook_form_alter().