Most law firms and law departments want an arbitration clause to govern should there be a dispute between them. The advantages are well-known.
Both sides, generally speaking, want the dispute kept quiet. Law firms generally like arbitration clauses because they think that arbitrators, who are lawyers, will be more sensitive to the need to make judgment calls and that a bad outcome doesn’t mean bad legal work. Arbitrations are theoretically cheaper and faster. Theoretically, because one downside of arbitration is that you are paying the arbitrator by the hour, and sometimes the process can drag on. But litigation can get out of control, too.
The downsides of arbitration are (from both sides) that you can’t compel attendance by third parties and the scope of discovery will probably be limited. But neither is usually a big issue with a legal malpractice claim – documents are few and available and there aren’t that many key witnesses, most of whom are the lawyers and the clients.