They surely would if they could, but since they will inevitably use it themselves they would need to collude and make a deal with the AI companies to find some way to keep the public out and law firms in (one example being targeted knee-capping of the free and low price tiers with respect to legal matters).
The ridiculous overcharging thing was something I thought the entire public was well versed in, so the idea that someone could earnestly think that they would get a discount seems absurd to me. As I recall, even the Oscars gift bags given to nominees in one of the years contained a prenup template supposedly crafted by best-in-business (divorce?) lawyers where you just needed to fill in the blanks. If you weren't getting a discount before AI, you are not getting a discount with AI I would presume.
It is understandable to see the moat as the illiteracy of the public regarding the legal system because the concerted effort to keep that illiteracy in place is a disgrace. But when the stakes are high, the moat still is finding a lawyer who knows their shit and one you can hold accountable. Which is why I find the idea of lawyers being dependent on AI much more terrifying.
Lawyers lobbying to outlaw AI-generated legal documents?