I have found a rock edge to be difficult to maintain, but yours looks like some sort of interlocking pieces (?) so may be easier. What I found with actual river rock is the grass just grows between the rocks making a rather messy look. The only place I've used landscape cloth is below a bark pathway. It works OK, but the weeds just colonize on the surface and eventually grow their roots through the cloth. Not enough to rip it out, and it does seem to deter weeds. Since this is a new home for you, I'd wait and see how things go this summer then decide what to do (if anything).
