What do you mean with you "don't seem to have any vertices that make sense"? In your second screenshot when you select all in Edit Mode, those two points make perfectly sense, they are the end points of that one curve - I actually used the same handles for the pen tool in Photoshop to draw the dashed line showing the curve:
And what do you mean they aren't there any more when you deselect them, at the ends of the curve are those black dots, those are the end points. Of course the handles are gone, they just show on selected points.
I guess your problem here is that the SVG is not imported as one complete curve but in many many small separated parts. This might not be the solution but that's all I can guess from the few information in your question. But to answer what you asked for, how come you don't have any vertices: you have them as I showed you. (Technically on a curve these are called control points, not vertices, but anyway.)