Good morning. Long time lurker, first time poster. I searched the forum for my issue and found some similar posts but nothing with a fix that I have not tried.
When I have it on heat, my 2012 Sahara is blowing cool air when idling but quickly gets very hot as soon as I accelerate.

When I first purchased the Jeep, I had a code for the thermostat so I replaced that. I do not know if the issue was present before this replacement because we bought it during the warmer months and I replaced ithe thermostat almost immediately. The thermo I replaced it with was one of the Duralast brands from Autozone.
I burped the system very thoroughly after replacing the thermostat suspecting that it might have air in the system but this made no difference in the heat situation whatsoever.
I also flushed the heater core suspecting a partial clog and it seemed to help last winter for the most part. My wife was driving it at the time so I can't be 100% sure that it fixed it then. Either way, the problem is back full force now. When I flushed it, I noticed very little sediment coming out.
I have no codes present.

I guess what I've narrowed it down to from my reading are the following questions:

1. Is it possible the new thermostat is bad? I don't have a digital temp readout but the needle stays in the middle as soon as it's warmed up and does not fluctuate when idling, at highway speeds, etc.
2. I do not remember which way I flushed the heater core. If I flushed it from the inlet hose out of the output hose, is it possible that it actually needed to be flushed the other way around? Unsure on the effectiveness of removing sediement when flushed in > out.
3. Could it be the water pump? It seems like the water pump would not be the issue since the temp stays steady once warm but I'm not 100% on this.
4. Something completely new that I haven't read about?

Thanks in advance for all your help!