I can select the nodes no matter if they are on top of another segment. The file is all segments and no closed paths. Open paths have no interior and can't be filled or colored. I don't understand what is being distorted. I think maybe you are asking about how to fill an area rather than how to select nodes with the shape tool. Normally, you can't color paths with no interior. You can attempt to use the smart fill tool to create new objects with interiors that can be colored using the segments as boundaries. I tried that here and got this. Maybe if we saw the file before you did the break apart operation it would be insightful.
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