All of your assumptions are wrong.
Also, your logic fails badly.
All that is necessary is to let the laundry build up until there is a complete load and then run it that night. In your scenario, there is some need to wait until the laundry has built up to require multiple loads. OK, you might need to have a few extra clothes (one day's extra) if the loads build up so that you need to run different types of loads on the same day.
There is no difference in this process if there is one person in the household or multiple. In fact, it is probably easier with multiple people in the household.
I suspect that the concept of "laundry day" has more to do with other scheduling issues than build up of dirty laundry.
But, go ahead, making up your wrong assumptions. Progress lies in the other direction, but you probably don't care.
Remember, that the occasional time the washer is run during the day doesn't invalidate the idea. The goal is shifting use of as much electricity as possible to night use and it's not a failure if some electricity is used during the day.
Perfection is the enemy of progress as well as the enemy of good