Advantages of mounting bottle under frame
Other?  I would not count protection as the first.  The cage is a lever and some edges.  A water bottle is not exactly an impact attenuator.  And you have the chain ring to get past.

You pretty much only see a third on touring bikes. It is the least convenient location but it is a third location.  Three bottles or a frame bag and one bottle.  Pretty much impossible to access while riding.  Typically don't have the clearance in the  bottom position for a long bottle.

Someone commented on pull downwards.  Notice I used solid loop cage below.

I give center of gravity zero points.  

On a mountain due to a small triangle it is typically the second cage so naturally it is going to be more common on mountain.  Still the least functional position. On a mountain bike you can shake out a water bottle. I tried it on my mountain and gave up. 

My best use of the last position is margaritas on Critical Mass. You give up volume to insulated containers. That is a large frame and that is 70lb dog.