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Off the top of my head, I can think of three reasons. The first is that often people want a more realistic shot rather than a high contrast shot (the contrast can always be increased afterward, but the reverse isn't always as easy). The second is that like any shooting mode, there are times when results are better without HDR, so enabling HDR by default isn't appropriate for many situations. The third is that HDR requires the shooter to be more attentive to how steady he/she holds the camera for a longer period of time, so if HDR was always enabled, a lot of folks would often end up with results where the pictures that were taken don't line up correctly.
You have the basics of what HDR does correct in your first post - it takes multiple shots at different exposure levels and then combines them in order to bring out both the lighter and darker levels. It doesn't necessarily increase the exposure time, but you still have to hold the device as still as possible (similar to how you would with a longer exposure) in order for each picture taken to be roughly the same for process of combining them. If you move while the various shots are taken, the software isn't always able to use the various portions of each picture properly because they're in slightly different areas with each picture.