I think that a button/checkbox to denote "Just remove duplicates" is the best approach.
Good points on the value of keeping these duplicates, though. For your research, duplicates marking stops may occur repeatedly at various locations. Places like overlooks, hilltops, etc, which may, for parks people, be predictors for park features like campsites, tables, etc.
I've been working on some code to process GPX files to do a few things I want. In the course of pondering duplicate points with timestamps, one solution is very easy. For a given set of points close enough to be duplicates, they are replaced with one point whose timestamp is the median time from all the points in that set. While this preserves total distance and time, it does mess up the adjacent segments' speeds...
Fun stuff!
Anyway I'll try both the simplify and network on this problem and see how well they work. (GPSBabel simply isn't working on this!)