Car starts without key!

Yes, unhook the switch so it's isolated. What I'm getting at is use a light bulb with a fairly heavy filament and a battery to test the continuity of the switch NOT an ohmeter

When you test something like a switch, which makes and breaks, and may be intermittent, using a test device ("big lamp") which puts more current through the test device helps to show up problems which the ohmeter might NOT show.

You have any junker bulb sockets, or can you get one? How about a headlight?