Successful update:
I replaced the temperature sensor, and now it works. It actually threw the E05 error once or twice shortly after replacement, then never again. I am unsure if there is a break-in period for oven sensors... [br]
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The sensor itself looked exactly like the old one. The wire connector was wrong (wrong shape and color), so I soldered the old one on with heatshrink wrap. I assume it is a resistor and therefore works in any direction. [br]
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I noticed that tapping on the white relay can be enough to turn off the heat or do some weird stuff (like possibly throw E05), and seems to be directly correlated with the start/cancel button.
Then I began testing.
I was able to cook toast (the default setting, medium, 5 minutes). Then, I did it again and then the E05 error was thrown. I unplugged the oven, held the power button for 15 seconds. I tried one or two other settings. I was holding my hand on the triad head sinks for the first few tests. They can get too hot to touch for 10-20ish seconds at a time. Shortly after, the E05 error was thrown again when I was tapping on the relay (seems to directly affect Start/Cancel).
I ran it on various high settings like largest frozen pizza, 20-minute roast, et cetera and it has not thrown the E05 error again.