Washing machines pull considerable power and the power scales with the water usage: They use warm water, either from a tap or using a heating element. Heating water is energy intensive, and the hot water quite literally goes down the drain. Current machines use about 50liter per cycle, heating that from 10 to 30 degrees consumes around 1.2kWh, to 60 it’s ~3 kWh. Doubling the amount of water obviously doubles the energy needed. A machine for a small family may see around 100 cycles per year, so the savings are substantial.

This doesn’t take into account the savings in drinkable water which has most likely been through an energy-intensive process as well.

Yet your washing machine does not need drinkable water specifically (can use rainwater), and your stats do not mention the energy/resource cost of manufacturing and shipping a new washing machine.

If you have collective washing machines as a public service running 24/7, the tradeoff may be worth it, but as a general rule it's certainly not the case. Extracting and refining materials for industrial processes consumes millions and millions of m³ of water and a lot of energy.

Our perception as consumers is skewed because the price we pay for electricity/water makes it economically more affordable to us to get newer devices, but from a resource usage perspective, that's certainly not a good deal for the planet. If we paid water/electricity the price that big industries do, we wouldn't even think getting a new device was a reasonable choice.

Washing machines consist mainly of materials that are easily recycled and very economical to do so. Metals, mainly, in a large package. Pure copper in the motor. All of which can be recycled to build a new washing machine.