It depends if you are trying to win the case if it gets to court or trying to force them to expedite the complaint/settle ahead of that. For many companies the cost of addressing the court action even if they win makes settling the pragmatic choice. Also legal threats tend to end up on the desk of someone who can do something, rather than an overworked contact centre person.
For instance I had a problem with damaged baggage that an airline’s complaints department had been sitting on for weeks. Within 24hrs of their UK registered office receiving a letter threatening them with small claims court I had a full settlement. They may have been able to prove their contract exempted them for liability if it had reached court, but they’d have spent more than the claim was worth even having their lawyers responding.
Sometimes it’s worth a roll of the dice.