The thing is, their goal wasn't to last for centuries. Their goal was just to not fall down, and because people back then didn't have the modeling abilities we have now, their solution was to be very conservative in their design and overbuild everything. If we wanted to make bridges that would last a long time, we could almost certainly do it more efficiently now.
Yes, stone will last longer, simply because it doesn't get eroded by the elements as quickly as possible. But it's also simply not usable as a construction material for bridges beyond a certain size and span, because it's only good under compression and even then its own weight starts to become a limiting factor as you get larger.
By the 1940s, a stone bridge that might take 100 man-years of labor to create could be replaced by a bailey bridge that goes into place overnight. I'm sure if we wanted to create a bridge that lasts 300 years, we could dedicate more resources to it, but it would be grossly wasteful to do so - you don't know how long the current requirements are going to even be relevant. It's better to build a bridge that lasts for 50-60 years, and have the option value of replacing it with a better structure down the road.