Another idea: when a takedown request is received, send the uploader a warning in advance of taking down the media, and allow immediate putback if the uploader posts a counter notice.
Counter-counter-notices are very rare as far as I can tell. People don’t file many spurious counter notices.
Somebody hosting a live event is probably sitting at the computer, ready to act on a warning.
The DMCA’s time limit is pretty fuzzy. It just says “expeditious”. So the site has enough time to act on a takedown request by sending the uploader a warning of impending takedown.