Thanks to Kyle and his post, I was made aware of a plugin called Widget Logic. This was exactly what I needed to use in order to get a couple of sidebar widgets to not show on the Events page.
Unfortunately, the Dynamic Widgets plugin just would not work with the admittedly old WordPress theme I was using and I'd all but given up when I found Kyle's post about the very same problem. After a little bit more digging, problem solved. Here you go!