Is there a way to get reminders on iPhone to persist until they're checked?
This is a problem os-wide in iOS, for other apps too, like calendars.  Once you have received your alarm notification, whether for a reminder, or a calendar alert etc, you're done.  There's no way to "snooze" or "remind me later" for these functions.

The only way to get this functionality at present, is to use a replacement app.  One such app is Due, the following blurb is copied from their website:



PERSISTENT REMINDERS
WITH AUTO SNOOZE

Overdue reminders stay in sight until you're ready to act on them.

And with auto snooze, your todo list never gets stale.

By default, DUE repeatedly notifies you of overdue reminders every minute* with its auto snoozing feature until you mark them complete, reschedule them, or turn off their auto snooze.

*Auto snooze interval can be changed to every minute, every hour, or none at all. There are no finer auto snooze intervals on iOS due to system limitation.