Why flights dont pass the north pole to get the shortest way?

Some flights do go over the arctic, and the North Pole specifically. See the link in the source section for a map of some common trans-polar flight paths. Generally, flight between the US and Asia can use that route to get the shortest path. Flights between the US and Europe go over Canada and Greenland - not quite the north pole.

Lots do. But, each airplane has to have an ETOPS rating, which basically says "this plane must stay within x minutes of an airport big enough for the plane to land at". So for instance a flight with an ETOPS 180 rating must never be more than 3 hours away from an airport it can land at. This means that some aircraft have to deviate from the shortest path a bit.

Aircraft will also sometimes choose a longer route with a tailwind instead of a shorter route with a headwind.

Lots of flights go over the northern polar region. All flights between the US east coast and Japan, China, Korea, and Thailand do. There are probably many others too.