I apologize if the format's a bit screwy- I wrote this last night and then edited it on the train this morning, and I don't blog from work, so I may not have got all the formatting right.
The overwhelming likelihood is that producing more lawyers isn't driving costs down because lawyers are choosing to do things other than law.
Legal fees aren't really "fixed costs". Court fees are, but lawyer's rates (per page or per hour) are not. Maybe the HOURS are fixed, but the cost per hour is not fixed. It's probably a little bit "sticky", as in once you're paying someone a certain wage per hour it's hard to cut their pay, but we're talking about young lawyers entering the industry, and they can be offered lower wages from the outset. This should result in a reduction of rates for the use of junior lawyers, which should save people money.
If legal fees aren't dropping, law firms do actually compete with each other on the basis of profits, and barriers to entry for law firms aren't that high, then that means that you don't have a flood of lawyers entering the marketplace. Which means they're not doing law – maybe they prefer to remain unemployed than take a lower salary because they think the salaries will come back up, maybe they are taking jobs in consulting or finance or management, whatever… but structurally, the argument you lay out isn't economically consistent with competition among law firms