I'm not trying to bump this thread, but wanted to add my experience for the folks spending money and chasing overheating problems that find this thread from Google.
I had similar issues to OP, running "warm" (220s - 230s) basically all the time and would go into the 240s+ uphill, on ramps, overtaking, etc. but cooling off some in the city. I had spent hundreds throwing parts at this problem--replaced nearly the entire cooling system, but with aftermarket stuff since I had no luck finding a Mopar thermostat. I bought this Jeep used (one owner) at 59k miles and strangely, it already had a MotoRad thermostat on it. Probably should've kept looking but oh well..
I finally took it to a shop, they thought maybe a cracked or warped head, but they talked me into trying a Mopar thermostat and radiator cap. When I brought it in, both components were Autozone Duralast which were both repackaged MotoRad items.
The shop's diagnosis was that the MotoRad thermostat wasn't even starting to open until 203* and wasn't fully open all the way until 220*, at which point my Jeep was boiling over and unable to do a proper burp with the system open. According to the shop, the Mopar unit starts to open at 195* and is fully open at ~203*. Like OP, a genuine Mopar thermostat and rad cap seems to have fixed the issue completely.
Moral of the story, if you buy an aftermarket thermostat and it is labeled to START opening at 203* (and t-stat temp
specs are typically given at the "start to open" temperature), you're probably going to have overheating problems as the already marginal cooling system of the JK struggles to keep up and your T stat starts closing again as it cools down to the normal temp.
