@stephenw10
I feel SUPER dumb. I did a sloppy job by not turning off all of the routing and firewalling capabilities on the old router, and that is exactly what the problem was. As soon as I disabled all the things, casting worked. I'm pretty sure that this wasn't a routing thing though because I only have the LAN port of the wireless router plugged in (and DHCP would have failed), but this router had firewall services applied to the physical LAN ports which had never been used before. Guessing that the firewall services didn't kick in for WiFi clients (which is dumb) and that's why I didn't have problems in the past.
Thank you all for being so helpful and dealing with my ignorance!