So, tell me how to fix this problem and make my life far less miserable. Flush, clean water coming out, thermostat is pulled so we can fast cycle it. Do one last fill, circulate and drain with distilled. Then we go back in with coolant, system is new MOPAR thermostat, new pump, radiator, etc.
Start it, watch for any coolant level to drop with the cap off so we can top it off if need be. Right about the time the dash reads a bit under 210, it explodes 2 quarts of expensive coolant out, the fan sprays it fucking everywhere. Now coolant is low, haul ass and top it back off.
As soon as the engine gets near the same temp as before, rinse and repeat. Blows coolant out the filler neck, fan sprays it everywhere, we top it off and it will keep doing that until we get tired of it or run out of coolant to add.
So you and Mr. 4 Low get your little heads together and tell me how to fix that problem.
Having thought about it a little more, I do remember doing the first warmup with the cap ON, and last time I did it I tried with the cap off and made a big mess as you described. The downside of cap-on of course is you can't really see what's going on, so you just have to trust the system to do what it's supposed to (push the air out into the reservoir when it gets hot and suck coolant back in when it cools), and check on it through a couple of warm-cool cycles, which isn't really that conducive to a situation where you're trying to get vehicles in and out on a schedule.
I might spend more time on the front end as well - I've experimented with different ways, like fill slowly until the radiator is full, squeeze the upper radiator hose to get any air out, put more in the radiator when the level drops, and repeat until I can't make it drop anymore. I've done things like filling it to just below the level of the thermostat, before actually installing the thermostat and housing, and then filling the rest of the way, which may or may not fit into your sequence of operations. I do this once every couple of years and my LJ (which hasn't even seen 1,000 miles yet in 2023) so I can afford to be slower about it.