Integrating an ESPHome device with Home Assistant is simplicity itself. Here's a chart of the power readings from the washing machine (left, s31green) and the dryer (right, s31pink) for a laundry cycle.

The washing machine goes through a lot of different phases, so it's understandable to see all those ups and downs. The dryer, on the other hand, being heated with natural gas, should be fairly steady. It's just the tumbler motor and a tiny amount of control electronics. I'm not too sure what those spikes are about. They last 1-2 minutes.

The really good news is that both of the monitors report zero power consumption (to 5 decimal places) when the machines are off. That's pretty unambiguous. It's an improvement on my homebrew design, which always seemed to pick up a little noise that needed to be dealt with.