If you place too much faith in brand new parts always working out of the box it will bite you one day, I say this because of my own experiences. QC only makes sure that x% of parts making it to the consumer are virtually guaranteed working, there is always a percentage, no matter how small, that slip through and become the bad reviews you see online.
Vacuum purging the coolant system is the best and recommended way of doing it. Using the extra tall funnel thing mostly works but you can still end up with air pockets in the system. They may work out eventually, but a vacuum system is a sure bet. Rent a pressure testing system from Autozone or Oreilly and get a hand operated vacuum pump from Harbor Freight and connect the vacuum pump to the testing cap and pull a vacuum on the system. Not a lot of volume in the system especially once you have coolant in there so it's easy to pull enough vacuum by hand to purge the system.
If your coolant is getting over 220 degrees and continues to climb higher than that then a cooler thermostat is unlikely to fix the problem of it overheating (unless the thermostat you put in previously is bad) since once it's open, it's open, The reason it continues to climb past that is you have hit the limit of your cooling system's ability to shed the excess heat faster than your engine can produce it. Even if you start out at a colder open temperature odds are you will put the 160 thermostat in and all it will do is cause it to take a little bit longer to overheat, unless the issue was still a bad thermostat. If you're taking the thermostat out to replace it I suggest testing the old one anyway to make sure it was in fact working. Also be aware that running a colder thermostat than the engine is designed and tuned for can cause issues. You really should get the engine computer reprogrammed to properly utilize the cooler thermostat, especially when going down to a 160 degree tstat. The computer is programmed based on an expected operating temperature for it's fuel maps and if it doesn't ever reach that temperature it might even throw a code. All depends on how the engineers decided to handle it.
Summit has a good article about running colder thermostats:
https://help.summitracing.com/app/a...ur engine,will NOT solve overheating problems.
Basically unless you're putting it in a race car that has the ECU tuned to utilize the colder tstat you really shouldn't bother with them.
Thinking that you're not getting enough flow seems odd since it's not like you're the only one running a Hummer H3 Alpha in high temperature environments. We've got a handful of people over in the Middle East (read: hot, dry, HOT, DRY, sandstorm!) on the forum that haven't had the kind of issues you're having. There's also a fair number of H3's in Texas that run fine with stock H3's. This is especially true since it sounds like you're not even towing or anything severe duty like that when you overheat.
So something is clearly not working correctly on your truck and it's not because of how the stock system is designed. Either a part you replaced was replaced with a faulty part, or it is something you haven't replaced/tested yet. Have you tried blasting the heat to see if it brings the temperature down at all? I know it'd suck in the middle of summer but if your radiator isn't able to keep up you do have another mini radiator in the HVAC you can use to supplement cooling with, and the coolant is constantly flowing through the heater core regardless of what the thermostat is doing. I have had to do that on not only the H3 but other vehicles I have owned that were overheating in the dead of summer. It makes enough of a difference to be noticeable. If that works then you most likely have a problem with either your thermostat or it still could be your temperature sensor.
Honestly I still suggest verifying your coolant temperatures first with something other than the sensor on the head. Or at least test the sensor and compare it to a brand new one and then put the new sensor in. The sensor is cheap and fairly easy to replace. I would at least start there next. Right now you're making the assumption that the sensor is accurate without knowing for certain. Figure out the actual temperature compared to what the sensor is reading and then go from there.
You can also check the temperature of the coolant coming out of the engine vs the temperature of it going back in. Top hose is the engine outlet and bottom hose is the engine inlet. If there is a large difference between the two yet your engine is overheating still then I would wager it's the sensor. Reason being if the top hose is good and hot but the bottom hose is much cooler then you're clearly getting flow, otherwise they will likely be about the same temperature as each other do to conductive heating in the coolant and no flow through the radiator to cause the lower hose to be filled with cooled coolant instead of heated coolant from the engine.