It's not necessarily bandwidth or range. Some of it can just be the quality of the router implementation. I remember having lots of trouble with a Chromecast (discovery issues, losing connections and other issues) and I couldn't figure out what was going on, especially since I had a brand-spanking new high-power access point.
But I wasn't using my new access point as a router. For unrelated reasons, I replaced the 10+ year-old router with something newer and the Chromecast started working flawlessly even though I hadn't changed anything on the wireless side.
OK so it's not the WAP, but I still think we could differentiate between an ethernet switch upgrade (for local problems i.e. with content streamed from one device to another) and a router upgrade (for upstream problems: NAT issues maybe?). Routers have switch functionality built-in, but since you were using a really old one you might have tried sticking a cheap switch between the router and your local network before upgrading. Anyway, I think the blanket "You need to upgrade your router." advice is unwarranted.
But I wasn't using my new access point as a router. For unrelated reasons, I replaced the 10+ year-old router with something newer and the Chromecast started working flawlessly even though I hadn't changed anything on the wireless side.