I used stock JK wheels and tires for a few reasons.
1- I got both wheels and tires for free. You can find stock wheels anywhere because they aren't worth much when so many people are upgrading to an aftermarket rim. Tires are the rubi take off 255/75/17 which I also run on my rig. The ones I had mounted were half treads that a member of my club got as a extras when he bought new tires for his rig. The guy just wanted them out of his garage so he threw them in as well. You don't need brand new tread for a trailer. They wear so much less with the minimal weight they will support. When your rig tires are shot for wheeling there is still enough tread on them that you could run them on the trailer for a lot of miles without any issues.
2- Extra spares for trail runs.
3- Easy extra height for ground clearance.
I thought I was going to be saving money by using the same wheels/tires on my trailer but its probably a wash. I had to buy wheel spacers to clear the trailer hub (just a necessity with stock jeep wheels) which cost me about $100. Then with mount and balance I'm at about $140. I would think trailer tires are pretty cheap and you could probably end up with 3 for about the same cost.
One thing I noticed about your OP,
He said he would put the most basic road trailer rated tire on there
I built my trailer to go off road, I don't want to run a road tire while I'm bombing dirt roads or washes and boulder fields. I took my 85% finished trailer out last weekend for a shakedown. In one place we drove through the trail was fairly narrow and there was a big rock on the driver side. I saw it and made sure to drive around it with my rig, as I got to it with my trailer tire I saw that there was enough curve or whatever in the trial and I totally tagged the sidewall with the rock. I don't know that it would have messed up a road tire or trailer tire but I wouldn't have wanted to find out right there.
I guess I'd add a #4 on my list above, peace of mind that the tire is robust enough to handle rocks.