Can dysfunctional graduate admissions processes be improved?
Who decides how the application process is handled?  If it's decided in your department, then I'd think you have a good chance of making it more reasonable --- maybe not anything like MathPrograms that costs money, but at least keeping all the files in one place where you can review what you saw last week.  Maybe you can get a secretary to scan the files and make them available on-line to the appropriate faculty members.  If the decision is made at the college level, then you may have more work changing the process, but money may cease to be an issue; a big expense at the department level can look trivial to a dean, provided the dean understands how bad the current system is.  If you need to deal with a central Graduate School, the subject of "help" (in quotation marks) in your message, then all I can do is wish you good luck.