Russ - we are looking to buy a 2013 Edge soon and have researched the dickens out of everything regarding using it as a toad.
From what I've found on several forums/sites, the Blue Ox base plate in the top picture mounts high to accommodate the Edge Adaptive Cruise Control Sensor (the black box on lower part of grille opening, driver side), which is included in one of the packages for the Edge (302A I believe). This puts the towbar attachments about 27" above the ground and I believe a lot higher than most MH hitches...thus a drop down receiver will likely be needed to keep tow bar level. I think the attachments sticking out thru the chrome part of the grille are a bit obtrusive also...IMO.
We are not buying an Edge with the Adaptive Cruise Control specifically so we can use other base plates that attach lower down on the front of the car and closer to same height as MH hitches. And personally think they look better coming out at the lower part of grille....but just my opinion.
Finally, I did see a post somewhere (don't have a clue where as I've think I've been to the end of the internet and back on 2013 Edge towing) where one resourceful owner relocated the Sensor so he could use a baseplate that mounted lower.
BTW, here are the links for DEMCO and Blue Ox for 2011-later Edge - note differences in connector height above ground for each...
Hope this helps.