Some racks don't have a problem with slop in the receiver fit because they are not just pinned though the receiver - they are bolted into it. My rack (never used on a trailer, just the van an a motorhome) is not a good one; it is just an older version of this SportRack
. It has only a 1.25" stinger, but it avoids rattling by having a nut welded into it so it is held into the receiver by a bolt through the hole normally used for a pin. When in a 2" receiver, it is used with a plastic spacer to fill in the extra 3/4" in each direction. In the main photo on that SportRack webpage you can see it is in a 2" receiver tube with the spacer and bolt in place; the instructions
for this rack more clearly show the bolt setup (images 17 through 20).
I think that a slop-free connection in the receiver will be easier on the receiver, although I trust that the receiver tube itself is suitably strong anyway.
Even with zero play in the receiver mounting, I think the bracing straps are a good idea, due to play and flex in all of the other joints of the rack. I have not used straps with our motorhome, but there is no where to tie them to (unlike the readily accessible Escape bumper), and the motions of the back of a trailer might be even worse than the back of the motorhome.
With a trailer, I would probably install permanent eyebolts (or hooks, or loops) for convenience when using the straps... although I would rather avoid a rear-of-trailer rack, for all of the reasons discussed in other threads.