Select everything and click the "Expand" button in the Control Bar.
Or, simply use the Direct Selection Tool (the white arrow) rather than the Selection tool (the black arrow).
You can try selecting the paths with the Live Paint Selection Tool (Shift+L) if you want to keep the live paint function and work in this fashion. You could use the Direct Selection Tool. (this should allow you to select segments of live paint items) or try selecting all then Menu>Object>Live Paint>Expand.
You are using the Direct Select tool (white arrow). When you use that tool and click on just the path, just the path is selected, not the text. This lets you edit the fill or stroke of the path or scale the path without affecting the text. But it also means you can”t delete just the path. Either use the Selection Tool (black arrow) or click on the text instead of the path before you delete.