The right front signal does not work, Arrow indicator also off?

Hello, it's possible the multi-function switch is to blame, but it might also be a bad Ground on that right front circuit. You said the test light checked out ok, did it blink when you tested the socket with the turn signals on? If so, I think I know what's happening.
If your test light flashes during the socket test, but not with the bulb in, there might be a bad Ground for that Right Front turn signal, and current flow is having to travel a different route to ground, passing through the RF Turn signal filament, then through the Park Lamp filament, and then finally through the LF Park filament to get to ground. And this is why when you use your test light it works fine because the test light has a good ground. This is just something to check before replacing the multi-Function switch.
The 3rd diagram below shows what might be happening. Try running an aux ground on that black wire at the Right Front turn signal socket and see if it resolves your issue. If it doesn't then you can check the Function switch some more, but this should be a quick easy test. Bad grounds do strange things for turn signals, and I know replacing the multi-Function switch is not always an easy job.