@Robk13
Yes, the traffic must pass through the gateway when living the PC. The turn traffic, not necessary.
But, your scenario is not good because you have a router(layer3 device) between your PC and your Gateway, this is not usuall. Unless your router A here is performing only layer2 connectivity, your gateway should be connected to the switch.
But, answering your question, yes, exit traffic from PC will pass through the gaterway. Because the PC does not know how to send traffic to the internet, so, it send it to the gateway.
The return traffic is a different situation. When the traffic comes from the internet and hit the Router A towards PC1, if the Router A knows how to reach PC1, it will send the traffic to PC1 and not to gateway.
But again, this is a very bad scenario.