Why shouldn't all law schools regularly host real oral arguments?

As I was gearing up this new blog adventure, Dave Hoffman at Concurring Opinion reacted to hearing appellate arguments at Temple School of Law by suggesting here that states "stop building new appellate courtrooms, and have law schools (in effect) subsidize some of the costs of the court system in return for educational benefits."  Howard Bashman of How Appealing, who happened to be one of the appellate lawyers involved in the arguments at Temple, followed up here by suggesting  that one lesson law students in attendance should have drawn "is the extent to which appellate courts are forced to ration justice due to the crushing burden of an overwhelming caseload."

These two intriguing posts got me thinking that all law schools should regularly host real oral arguments — and, ideally, lots of different kinds of oral arguments involving different kinds of courts hearing different kinds of cases.  A very enriching and useful aspect of my clerking experience was observing oral arguments and discovering the wide diversity of judicial and litigant styles.  Now, as a law professor, I continue to learn a lot from reading SCOTUS oral argument transcripts and from watching/listening to on-line oral arguments in my field.

So, furthering Dave's idea, I would encourage law schools to actively recruit (local and distant) courts to conduct oral arguments within its walls.  Law professors could actively develop innovative educational programming around these arguments — by having students read briefs before the argument and perhaps even script questions for the judges; by having students draft opinions based on what they saw and perhaps even write reviews of the lawyers' and judges' performances.

Does anyone know if any law schools host real oral arguments on a consistent basis?  Does anyone see any real downsides to doing so?