Before you remove the switch or the starter, I recommend a couple of test.
When the car will not start, remove the T12 connector on the relay board (the one that goes to the starter, backup lights etc). Then carefully pry the top black piece that covers the wires. The wires are likely to pop out of the connector, so just take your time and go slow. Place the connector back onto the relay board and push the wires in.
Since the cover of the connector is off, you should be able to see a large yellow wire, this should go to the starter. There is a red wire next to the yellow wire. Run a jumper wire from the red to the yellow. The starter should engage. This should help identify if the problem is upstream or down stream of the connector.
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