If they followed the DMCA instead of their own system it would scale just fine.
It would scale because under the DMCA system the site is not responsible for finding violations. That is up to the copyright owners.
All the site has to do is:
1. Take down alleged infringing content when someone claiming to be the copyright owner files a take-down notice.
2. Put the content back when whoever uploaded it files a counter-notice.
3. Tell the former that if they want the content taken down again, sue the latter. The site is now in the DMCA safe harbor.
All the site has to scale up to handle is dealing with notices and counter-notices. For that, all they need to do is check that all the required fields in the forms are filled out. This does not require anyone with any legal training--it is just checking things like they have identified themselves, described what content they want taken down/put back up, stated a reason for their belief that this action should be taken, and similar things.
You could train in an afternoon anyone who can read at a pre-high school level to handle this. 20 people in a normal shift could handle those 7200 cases per day.
Heck, you could even speed that up if you wanted by doing even less review on the notices. There aren't really any legal consequences to the site if they accept a notice that wasn't quite right. It is only the counter-notice that needs a little scrutiny. You want to make sure everything is correctly filled out in that, because it is the counter-notice that gives you the safe harbor.