The door to my house is the door of my house and the way to enter my house. This is the road to the beach. This is the road to take to get to the beach.
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MrPedanticI would agree with CJ here: "to" = "that has a functional relationship with".Yes, of course, it's the "belong/of" meaning, but IMO, in the original context, when one opens to door to the house, it also means he/she opens the direction/path a
"The key to the door"; "the door to the house"; "the clue to the murder".
MrP