Removing Couch Grass
Couch grass is a common and very invasive garden weed and rapidly spreads by rhizomes underground which are very difficult to remove especially on clay soil as you are finding out.
A lot of allotment holders have recommended planting the area with turnip seed which seems to work very successfully as the two plants do not mix and the couch grass then withers and dies back.
You are right in that any small bits of rhizome left in the soil will regrow to form another network of rhizomes. The longer the length of a rhizome piece left in the ground the more likely it is to regenerate and the greater the depth it will emerge from. More shoots are produced when rhizomes are cut into smaller pieces but these are less vigorous and often die off.
There is less chance of the smaller fine roots growing back as they are less vigorous and they will die before reaching the surface.
I would try to do the best job possible and if any do survive dig them out before they have a chance to grow bigger and spread too far.