That 250 mile circle is one of multiple "spots" viewed by each satellite. Each satellite can see (IIRC, from a post above) some 2300 mile diameter circle, which is covered by multiple "spots".
Your principle of being able to track is okay, the satellite network knows which spot(s) are getting signal, and can track signal motion across the face of the earth. Just a nitpick about how many 250 mile diameter spots are involved (many more than 60).
You are correct to a point. If you ever used an Iridium phone you will loose the link from time to time. That only says that we may have missed some of the pings. However, 5 or 6 pings is about the right number to reconstruct a ground track.