The tank could easily be leaking. Such leaks often start small and get bigger and it would be enough to extinguish the pilot. There is no real rule of thumb for tank life. Six years could be very little in some places or quite a lot in others.
If it is condensation, it should stop fairly quickly after the tank fires up. If it does it all the time, it’s a leak.
By the way, $400 in labor charges is not insane. Around here, that is rock-bottom cheap. High end can be as much as $1,500. An installation begins when the plumber leaves his office and ends when he drops your old water heater at the dump. In many places, he is also required to get a permit from the city.