Not sure I understand the question - if you use Insert Line Segment command - that can add vertices on any linestring and can do it on the line (use Near Snap to do that for example) or offset to the line to change its path.
I am not sure what you mean when the linestring is connected? do you mean when you join two linestrings together you want a vertex at the join location - I think we do that today - it has to do that because there may be an angle break in the line at the join point? I think I am missing something here - I have no idea also what the AutoCAD image is trying to show me?
The "After The Fact" Line Editing on Multi Segments - that is a command I think - adding that into Linestring Editor would make that more complex ta it is right now and I think lends itself to a small dashboard command where you can select single or multi segment (from here to here) and then change the property of the linestring etc.
We can look into that at some point.
Alan