A huge pile of escrowed cash, held against any release of records (inadvertent or otherwise), does seem to be a way of moving some of that trust around.

The only way I could think to do it would be to have it process the transaction in memory and never store the data at all. Make the running code open source, and make it hashable in some way so that when you visit the site, you could tell if the current running version is the same as the one in the code repository.