It looks like you're using the Gcodetools extensions. Then switch to the Selection tool and deselect everything, And then you should be able to select the part of the path that you don't want, and delete it. Then select the new nodes, and click Break Path at Selected Nodes button, on the Node tool control bar.

You can do that by double-clicking on the path with the Node tool (after you already click on it once, to select it). If there is not already a node on either side of the part you want to delete, you'll need to add new nodes. If that's the case, you'll need to use the Node tool. Unless you mean you want to delete part of a path? So you should be able to delete whichever part you want - the larger or smaller path.

You can see it by selecting the larger path with Selection tool, and looking at the status bar. Although I'm not sure if it comes from tracing, in this case. So what you said is true for some options, but not for all options. I've noticed lately, that using certain options, the results are not grouped. In the past, every result from Trace Bitmap was grouped. Depending on the image being traced and the options used, the results might be grouped. At least not with Inkscape's Trace Bitmap. It's not a group of objects, so ungrouping won't work. Tracing a bitmap produces one single path (which may be made up of multiple sub-paths).

Bitmap to vector image inkscape