A suggestion and some thoughts...
1. I think it would be worth it to remove the thermostat and see if the car still overheats. If so, we'll know it's not the thermostat. If not, you'll know to get a new thermostat and make sure it is installed in the correct direction.
2. Water pumps sometimes are sold in two versions, each for opposite engine rotation direction. If you somehow ended up with one that turns in the wrong direction, it would still pump, but it would pump at a very low rate.
3. I haven't heard of a clogged radiator on an LS, but it certainly has happened on other cars.
The reason it isn't overheating with the heater on is that the heater core is like a mini-radiator. This suggests that either your radiator isn't working, or coolant isn't flowing to it. Is your hydraulic fan working?