When I did research on this topic, I found four camps and only one appealed to me:
1) The Frame X folks
2) The Frame non-X folks
3) The Axle X folks
4) The Axle non-X folks
The non-X folks would raise concerns that if one of the straps let loose, you have the other side of the X that would want to pull the vehicle across the trailer. That seemed logical to me, so I avoid the X styles.
The Frame folks argued for the reduced body roll. On the trip to Moab with Doug, he stated the need to pull the frame down to the bump stops. That sure did not appeal to me, since it takes a LOT of tension to make that happen...and I saw his come loose frequently (almost every fuel stop).
I do know that some axle strap folks will use smaller straps to frame (bumpers, whatever) to reduce body roll while still using the axle straps to keep the jeep on the trailer. I think Todd and Justin do this, but I could be mistaken.
Airing up the tires will help avoid what Don mentioned/experienced, since the low air pressure allows some give in the tires and pressurized tires will not have as much give/flex.
I go for the axle in a non-X form with no concern about body roll. I do not air up my tires, my straps are the basic ones you get at Lowes (although I will likely upgrade to the safety hook style at some point), and I have never had a problem. Of course, i have not jumped a curb yet, but I have experienced plenty of rough roads and pot holes. I have seen the MAC straps go slack on Wes' setup...he blamed cold weather (maybe rain) or something like that...I really don't know what happened, but it made me question if the material they use has some stretch in it versus what my cheapo Lowe's straps have.
I do have 4 cargo ratchet binders and chain, but I was too inept to get them to function with axle straps (they just twisted around and around...very frustrating considering how much money I have in them). I know Rob uses chain and ratchet binders successfully on his setup.