As of Cascade CMS v8.2 the Start Date field publishes the asset at the selected date. If a user receives the following error during submission, that means their Site Role does not allow them to Publish:
You are not authorized to schedule future publish dates. Please contact an administrator for assistance regarding your publishing rights or send the changes to an applicable Workflow.
You can verify by checking out the user's effective abilities under Publishing, specifically Publish writeable Home area assets. In order to set a Start Date in the future, the user's Site Role will need to have that ability enabled OR be assigned another Site Role with that ability.
Keep in mind that enabling that ability will allow Users with that Site Role to publish assets they have write access to.