Yes I know that feeling and garden on clay and had docks, though less perhaps than you. Maybe you should aim to reclaim a small corner first and do it really well to inspire people.
Based on your description and in an ideal world I would buy an http://www.implementations.co.uk spade (sharp, easy and quick to use) and pay somebody to remove docks from part of the plot, say 3 allotments worth. Cutting the top 4-6" of roots has worked for me. Then mulch with 6" approx. of manure, grow potato, courgette, squash; not salad or carrots in first year.
I think that topping the docks does not solve the problem, just stops them seeding, and organic matter will improve drainage withing a year or two so no need to mole plough.