Are auto-apply SuitePromotions supposed to appear automatically in SCA? At what point does it check for those promotions? I have been trying to test one and I can manually enter the coupon code, but it never auto-applies itself otherwise when adding/editing the cart.

There shouldn’t be anything ‘eventual’ about it. It should auto apply whenever a recalculation of the cart happens, eg: 1. Item added/removed 2. Item quantity changed 3. A (different) promotion applied, etc