I'm confused because, if an element is just a resistor, opening it would
just cause it to stop heating. But even with the oven switch turned off,
the fire was still going until the fire department closed the oven door
and
shut off the electricity to the house.

That is a darn good question. Turning the switch to off should have killed
the power just as effectively as shutting off power to the house. It did
for me when my broiler element started arcing a few months ago.

You probably only thought you had the switch off... But if you are certain
you did, then you have a bigger problem than a failed element, and it might
be prudent to replace it.

Just in general, it would be a good idea to have a fire extinquisher in the
house. We have 4. Never used them.