Regarding the issue that a Fuji camera will always be in TTL mode if a Godox product is in the hot-shoe:
I got a sad response from Godox, but at least we know now that this issue will not be fixed. From this point on I can not regard Fujifilm and Godox to be actually compatible. (Many users will off course never feel this is a problem, but for shooting very timing specific action (think skateboarding or dancing) a delay to the shutter (because camera is always in TTL) is no joke.)
Here is the message from Godox:
Since it works for you with a Nissin flash, there is definitely hope. The first thing I would try, would be to cover all connectors in the hotshoe except the big center one, with a piece of tape. This makes your camera act like there is a "dumb" flash connected, e.g. one that does not do TTL and other fancy stuff (like your Nissin).
The sad thing is: In many ways, Godox does not make their products truely compatible with Fuji. When you put a Godox flash or trigger on a Fuji cam, the camera is forced to use TTL which is NOT a good thing. Essentially one can only escape TTL by hardware hacks (like the above tape) or by using the camera function assigned to an Fn button: TTL Lock. (I am not sure all Fuji cams has this - my X-H2 does.)
(Btw: Using PC sync cable should work the same as using only the hotshoe center pin, so you might have had a bad cable connection. Those PC sync connectors are not exactly the most reliable connectors in the world.)
Oh, and yes, one can put your Godox flash in manual mode, but the point is that your Fuji camera still behaves like it is shooting TTL, which creates problems as the above, as well as introducing shutter delay and making it impossible to shoot in Fast Continuous mode with flash.
On X-H2 you can assign any Fn button to be a toggle to bring you to C1 (Custom setting 1). So save your desired settings as C1 and assign an Fn button of your choice. (Asuming that X-H1 also has this function. If not, your option will still be to use one of the C settings, and then just use the top dial to go to it.