Does the browser need to be launched with the global proxy to work properly?

If you’ve set a global proxy, the browser should pick it up without needing to be launched with the global proxy explicitly. Chrome typically follows the system proxy, but Firefox can be a bit different since it has its own proxy settings.

I’d recommend using a tool like Charles Proxy or Fiddler to test if your proxy is working as expected. They’re great for debugging and seeing exactly where your traffic is going.