Modern dish washers use under 4 gallons per cycle, so given you have at least one wash and one rise cycle, you’d only use a about 50% more if you are able to opt for rinse only.
Compare that to a 2.2 gallon per minute water/efficient kitchen faucet where you’ll approach the amount of water consumed by your triple cycle load after just 5 minutes of leaving your water running.
5 minutes of running water at full speed to wash up dishes seems like way too much, unless it's a really large household. Washing up after a quick meal takes a few seconds, and if a proper dinner with cooking you'd fill the sink enough to soak the load then come back to rinse.
Running the water continuously is not recommended but for people who were not taught water-saving practices, it is likely the default way one might go about hand-washing dishes.
My data was from memory as of >10 years ago from when I worked in construction, I am very happy to hear it is likely even less than I originally thought.
Compare that to a 2.2 gallon per minute water/efficient kitchen faucet where you’ll approach the amount of water consumed by your triple cycle load after just 5 minutes of leaving your water running.