Firstly, you can't wash with cold water because soap doesn't activate with cold water, hot water also kills bacteria and helps cut through grease.
Fats themselves are hydrophobic and without activated soap you wont get them off... enjoy your "filmy" dishes.
Second, humans expend a lot more energy than you think. The act of standing and using our arms releases varying amounts depending on physical fitness but averages somewhere in the 1kg/h ballpark. https://www.jstage.jst.go.jp/article/jjphysiol/50/2/50_2_199...
Dishwashers use about 1800 Watts and are commonly run for 30 minutes, the average co2 in the USA is 0.92lbs per kWh.
Meaning it's _basically_ the same.
Then there's the freshwater usage, which is the real kicker, because dishwashers use significantly less freshwater, and freshwater filtering is the largest environmental impact of washing dishes (not the direct co2 output).