Kenmore HE3 seems to use way more than 10.5 gallons

On average, washing machines use 19 gallons of water per load, and the average household runs between 5 and 6 loads per week. Based on those figures, most washers use up to 5,605 gallons of water annually.
Front-load washers use an average of 12.5 gallons of water per load, while top-load washers average 19.6 gallons per load. Older top-load models manufactured decades ago use 40 or more gallons of water per load.

Possibly reading the meter incorrectly...

ck here...
https://plumbingsniper.com/how-to-read-a-water-meter/

My 1.5" water line to fill my pool puts out 12 gallons a minute
it would take 45 minutes running full out to put out 530 gallons.

Best to you.

Conventional top-loading washing machines with center post agitators use about 40 gallons of water per load. Compare that to front-loading machines that use somewhere in the neighborhood of 20 gallons. Remember, that 40 gallons will require twice the energy to heat, too, which is one of the key costs of operating a washing machine.

https://home.howstuffworks.com/washing-machine-water-usage1.htm

https://prudentreviews.com/how-much-water-does-a-washing-machine-use/

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maybe this will help

8 cuft3.

If there is only hardness (calcium/magnesium) and NO iron you can set the hardness at 8. It is best to round up though to 10. This sets the amount of gallons before regeneration and lbs of salt per regeneration.
RJH2O

This is from not enough HOT loads.

Fatty deposits gather between the basket and tub during COLD washes.

Then they are being released in large numbers when you do your whites on HOT.

It is recommended to do 1 HOT load per week.

Do an empty load with some degreaser added (like Goo-Gone) on the fullest/hottest setting you have.

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When set on large load, it will use close to 30 gallons of water. On the smallest load setting, it will use right around 20 gallons. This is what makes the new front load washers look so good. They average around 4 - 6 gallons per load. Have a great day.

the inlet valves are shot. what happens the coils in the inlet vales can no longer hold the water from going into the unit, causing water to run into unit when not in use. replace the inlet valves and you should be good to go. the inlet valves can be removed by, removing top panel(3screws in back), disconnecting hoses, taking the small clamp on the small hose connected to the dispenser. theirs a small screw holding the inlet valves from back pane, remove old inlet valves and replace and you should be all set. hope it helps.

may see some but not much..front loads only use 4-6 gallons and is adj as water is absorbed by clothes ..a typical top load will use 25 gallons or more per fill thats 50-55 gallons per wash verses 25-30 for front loads also you should see very few suds ..be sure to use HE detergents to prevent oversudsing