There's quite a lot in this forum about the idiosyncrasies of the start/stop system in the MK3, you might like to try a search.
Summary: there is a very long list of factors that will inhibit it (check the handbook), and it's quite difficult to diagnose which one of those is screwing it on any particular occasion. Not helped by the dashboard display which may not show a helpful indication. If you do lots of relatively short trips, the car may not be happy with the charge state of the battery (do you get a battery symbol on the right-hand display?). My own personal experience is that it hardly works at all during the winter months, but normal service resumes as the weather warms.
Regarding your description of the dashboard lights. I think the comment about the indicator coming on when you turn the engine on refers to the initial display of all the various warnings, it'll be one of the many lights that come on as the car checks the various systems. If you're certain it's not there, that suggests a system fault that might need a dealer visit to diagnose. After starting the engine, stop/start is enabled by default ... although may not actually work. When you push the button, you are turning the system off, and the warning light comes on to remind you that it's off.