I've done an Atlantic crossing with an item that required us to go non-ETOPS (I don't recall what it was at this point), and yes, we had to stay within 60 minutes of a diversion airport. However, unlike ETOPS-approved airports under the applicable op spec, which are checked for suitability during dispatch, the 60 minute diversion airports don't have to be anything more than a piece of pavement that can handle the jet. In other words, there's no requirement to check closures, weather, CFR capability, etc. for non-ETOPS diverts of less than 60 minutes. While this certainly isn't a problem flying over civilized areas, it's a good idea for the crew to do a bit of checking themselves before launching over northern Canada or Greenland on a non-ETOPS "Blue Spruce" route.