It'll be like software. The demand for both software and legal services is effectively infinite. If it becomes cheaper to file lawsuits then we'll get more lawsuits, not the same number of lawsuits at a lower price.
There will still be jobs for competent associate attorneys to prompt LLMs and review output. But frankly a lot of attorneys — especially many of those who attended "Third Tier Toilet" (TTT) law schools — were always a waste of money and should have never entered the profession in the first place.
There is a huge problem in at least family law where a tactic is to simply ddos your opponent with motions and frivolous complaints until someone runs out of money and settles. I'm watching someone finally successfully defend themselves using a Harvey like tool they built themselves. An appellate attorney they consulted with recently, told them they had never seen anyone going pro se get as far as they had in appellate court. So I'm hoping that this serves as a return to making the legal system more accessible, and not just something used to railroad people.
You can’t 10x the number of lawsuits, as the courts are already at capacity. If frivolous lawsuits become common they’ll just change the laws to require a greater damages for it to be heard.
Lawsuits are just one type of legal services. There will also be more contract reviews, more cease-and-desist letters, more patent applications, more regulatory comments, more license applications, etc. Although I do expect legislatures will eventually budget for expanding civil court systems in response to lobbying from business groups to speed up trials.