The lawyers have figured out that the more laws, the more need for lawyers. When the gov't pays both sides, some desk jockey will figure out that the more complicated the law, the easier it will be to ensure that the gov't wins every dispute as it gives the gov't lawyers (both sides being paid by the gov't) more wiggle room to let the gov't win.
Universal legal coverage will mean that the government pays your lawyer to explain to you how to bend over to make it hurt less.
Just as the government wants to limit the amount of money a physician can charge for a procedure, so it should limit the cost per hour for legal services. If you have a right to the services of a physician, you certainly are entitled to a competent “navigator” with respect to dealing with arcane language and indecipherable legal boilerplate.