Hi Shane, sorry it's taken me a while to get back, I took the day off on Friday!
Regarding correction of undercutting by vertical down welding, I guess it really doesn't matter what my opinion is since the Code(s) generally permit it. But, since you asked, I guess my observation has been that a fairly large percentage of undercutting I've seen through the years has been relatively rounded at the very bottom of the undercutting, thus reducing the stress notch effect. I remember some years back, the Naval Joining Center did a study of intentionally applying fairly shallow undercut with a rounded profile at the toe of fillet welds; tests showed that this actually worked to enhance the "strength" of the welds. (Naturally, this was done mechanically rather than thermally to be able to control the amount and shape).
From my recollection, the correction by vertical down (at least in D1.1)stemmed from one old timer Committee Member who was a pipeliner and a major player on one of the large producers of filler metal. In fact, if you look at the Lincoln Welding Manual, you may still see vertical down referenced quite often.
As for the mechanical effect, I guess unless the undercutting is gross, my opinion would be the vertical down is more or less a cosmetic fix. So, for what its worth, there's my 2 cents!
By the way, I don't remember as far as D1.1 goes but ASME IX allows the welder who was qualified up to weld down for repair of undercut or weld root when the root is to be backgouged.