I ran into the same issue, and it took me a while to stumble onto an easier solution than making duplicate components without the silk screen or editing the Gerber file.
Right-Click on the component that you want to remove the silk screen from, and then from the drop-down menu, select Ungroup Component. Then you are free to select thee individual pieces of the component to delete, move, or otherwise edit as you wish. You can select individual lines, arcs, pads, etc.
A caveat, though, is that since you just destroyed the component's identity as a component, the RefDes, Value, or other properties that you normally want, are gone, and will have to be placed back onto the silk layer individually by placing text.
Hope this helps until a real solution can be put into Diptrace.
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