Saving your file using "File>Save As>Optimized SVG" avoided the black shapes, anyway it only shows one gradient so maybe you scaled the other paths after the gradient was applied, maybe you didn't, who knows.
As workaround you need to reaply the gradient, make solid color the path you have problems with, later apply a gradient, if you need to apply the same gradient to another path, see this
post.
Check it out in the gradients menu that all of them are unique, the # column shows how many times that gradient is used, if you see a 2 or a bigger number you'll have problems in GTS, the gradients # column should look like this one (marked in green)
Once you are done with the gradients and want to upload your file to GTS, first clean up your document, you can find it at "File>Clean up document", and now the most important, in order to be succesful, save your file using "save as optimized SVG", you can find it at "File>Save As>Optimized SVG", use these settings:
I see you used mm for the canvas size units and a scale different to 1, try to avoid that and use these page settings in red, you can find them at "File>Document Properties>Page
You can use 512x512 for square view, 1024x512 for landscape or 512x1024 for portrait orientation, these sizes are not mandatory but they are recommendable.
One last note, part of one of your paths is outside the canvas, you won't see in GTS anything that is outside the canvas.
I've attached the working file in a zip.