I was trying to maintain footing in an outdoor last winter that was (supposedly) river sand. All the fines had washed away, and what was left was too large and too round (except for the actual rocks or chunks of gravel, but I digress). It needed to be WET, like, it rained so hard it was a total lake and loose yesterday, today it’s perfect. I actually sent the sand off to a footing company to be analyzed (ring was not built by current BO), and they said it’s all off the charts on the large side, you can’t add anything to it to firm it up except to replace most of it with different sand first.
So, based on these pictures, it might feel loose to you because it’s the wrong kind of sand for an arena.
And all that said, I either dragged with the tractor or we pulled a chain drag with a Gator. If the footing was actually wet enough to be useful, an ATV would overheat (but it also had other problems to be fair). We tried at one point to pull the heavier pasture drag chain (flat side down) with a pickup to see if the weight would help with compaction, and the pickup struggled without 4wd. I never had to put the Gator or the tractor in 4wd.
I think a Jeep would be a reasonable try. You’ll want to go slower so not to dig in the corners too much with your tires. If the looseness of the footing gives you trouble, use 4wd high, but be mindful this will also cause your tires to cut in more, and it may be counterproductive. If the sand is on the drier side, you should have less resistance and weight to drag against than when it’s wet, so your vehicle won’t work so hard.
But I don’t think that this kind of drag is going to be the way to go. The chain drags do tend to promote the larger particles sitting on top. Ideally, you’ll want a real arena drag with a roller on it, which probably means pulled with a tractor because of the type of hitch used.