One twin getting inside and their erstwhile companion starting the cycle is absolutely possible.

In the animal kingdom, the stronger siblings will often murder the weaker ones to assert their superior evolutionary fitness. It's best not to interfere and allow nature to take its best course.

There are custom-made products for this. You don't have to buy from them obviously, but search Amazon for baby proof washing machine. Should be under USD20.

Internet-connected plug?

It allows for custom auto-off timer, or you could just manually turn it off via an app on your phone.

Baby gates. Baby gates everywhere.

Also thats a weird toddler lock. Ours you have to push and hold a button to activate it and it wont run until you push and hold something else.

Can you remove their access to the room where the washing machine is? Mine is in the attic, behind a door and a stair gate, so they couldn't even come close.

What brand are your machines? Because the child lockout function you mentioned absolutely should prevent a cycle from being started. That's what it's for.

Really, that hard to pull a plug when you're done? This is a non-issue. The plug too far? Okay, get an extension cord.

I know this maybe sounds like an easily solvable problem - just turn the power off when it's done - but that's just not how things roll in our house.

Maybe try fixing your house where pulling a cord to keep toddlers safe takes priority over the inconvenience of pluging something back in.