Your potato experience has happened to me before and couch grass is difficult to eradicate in ornamental flower borders because of clump infiltration from the couch roots.
Yet despite its thuggish appearance it is actually one of the easiest perennial weeds to control on allotments or vegetable patches. The roots don't go down that far, no more that a forks depth and they, pull up easily and are easy to see. A methodical approach to clearing ground usually does the trick.
So often new plot holders have the idea of rotavating their couch, bindweed, creeping thistle infested plots, attempting to save work by clearing ground quickly yet all they really do is propagate these perennial weeds by chopping them up into smaller pieces potentially making the problem worse not better.
Couch grass is also highly sensitive to Glyphosate.
Growing potatoes to break up rough ground and help clear weeds is a good idea as long as you don't expect 100% top notch crops.