Try enabling that option, but set it for a different time. See if it then sets that alert as well as the "00:00" alert for new items.
When setting the iCloud.com alert setting to "1 hour before", events created via iCloud.com get that single alert attached (showing on all devices). Events created in Calendar on the Mac still only get the midnight alert set.
Thing is, when creating an event in Calendar on the Mac, no alerts are visible there. iCloud.com also shows no alerts for such events. It's only on the iPhone and iPad the rogue alerts are showing.