Advertising is one thing. Advertising in a court filing is a whole different deal. Lawyers are expected to advocate for their client as hard as they can, but they are expected to stay within the bounds of the truth while they do. ("Duty of candor" is written into the rules of most or all jurisdictions, at least in the US.)

If lawyers are going to straight-up lie, then that can cost them the case, or get them sanctions, up to being disbarred from practicing.