Alright, this is weird. I ran tcpdump on the server, and checked both physical and wg0 interface. For things like youtube, it's a constant stream of packets coming in on the physical interface, then immediately being relayed through wg0 - just as it should be.
But for Crunchyroll, there's.... Nothing. I get an initial burst of packets when opening the site containing the video I want to stream, and then packets just stop coming in once the page itself has fully loaded.
Awesome. Thanks.