This answer to my own question is the best example of which I know that seems to show that the maximally extended Schwarzschild spacetime is of some interest other than as a purely mathematical object. This is based on a physics.SE [answer][1] by Ted Bunn, which was a summary of a longer [article][2] he and Matthew McIrvin wrote. Theirs was about a Reissner-Nordstrøm spacetime and dealt with electric charge, but I've modified it (hopefully in a correct way) to relate to the less esoteric and more physically realistic Schwarzschild spacetime. I would be interested if other people could provide better motivation for the notion of maximal extension than what I've given here. 

People often ask how the gravitational field of a black hole can escape out past the event horizon, or, in a more sophisticated formulation, how the information about its mass can get out. For a black hole that forms by physical collapse, the answer is that there is information in an external observer's past light cone about how much mass fell in -- this information comes from the mass that went into the gravitational collapse.

But the Schwarzschild spacetime is a lot simpler than one formed by gravitational collapse, and we would like to have a valid answer for it as well. The answer here is that we form the maximal extension, which includes a white hole singularity. The information about the mass is on the observer's past light cone, which contains this singularity.

  [1]: http://physics.stackexchange.com/a/12171/4552
  [2]: https://facultystaff.richmond.edu/~ebunn/ajpans/ajpans.html