Well it's been a month, but I was finally able to get back to the suspected blown element on my water heater. I drained the tank and pulled the lower heating element - it's definitely blown. See photo here:
https://imgur.com/a/fJ7wxRN
I changed the element, filled the unit with water and turned the power back on. An hour later, I had hot water and the breaker hadn't tripped or anything weird like that. Great! I walked away to spend the evening with my family. But an hour later, I was outside and noticed that hot water was gushing out of the temp / pressure relief valve pipe (we have the TPR routed to a pipe outside the building, so hot water was streaming down the side of the building). I shut off the water and the breaker and went and poked around the water heater and noticed that the hot water pipes seemed really hot, and the water in the faucet was also really hot. I was out of steam and decided to try to resume the battle the next day.
18 hours later, I went back and measured the temp of the water in the tank - it was 140 degrees F. This was after sitting with no power for 18 hours, so it was probably boiling (or close to it) when it tripped the temp / pressure relief valve. Both thermostats on the water heater were set to 120 degrees. At this point I concluded that the thermostat must be busted as well. Maybe this was the problem the whole time and I was too stubborn to consider the likelihood that the thermostat on a brand new unit was faulty (but since the old water heater exhibited similar behavior, I'm not totally convinced yet).
Anyways, I have now replaced both top and bottom thermostats and am monitoring - so far the water is heating to 120 degrees as expected, so that's good. But I'm going to continue to watch and see what happens.
One thing that I would like to clarify - I used my clamp meter to measure the red and black wires at the panel after I replaced the thermostats and turned the power back on. Both the black and red wires were measuring about 19 amps during the heating cycle (this unit has 2x 4500 watt elements). Should both wires be sending current during the heating cycle, and does 19 amps per wire seem about right?
Thanks, and happy new year, everyone!