Selfhosted

When you "cast" from the phone to the Chromecast, what happens it that the phone gives the Chromecast an URL where it can find the stream to play. The phone can read that stream because it's connected to the Tailscale VPN. The Chromecast isn't, so the stream URL is inaccessible to it. You see the Jellyfin logo because that's a feature of casting (the app on the phone gives the Chromecast a logo to show).

There's no point in announcing the subnet from your laptop, because your laptop is not a router for the local LAN. You can use this to reach local LAN devices from remote Tailscale nodes but not the other way around.

Some possible solutions:

1. You enable the hotspot feature on the phone while connected to Tailscale, and connect the Chromecast to the wifi of the phone. But the stream will consume the cellular connection, because you're using the phone's wifi for the hotspot (the phone only has one wifi interface so it cannot use it both to connect to the local LAN and for hotspot).
2. If you connect the local laptop to the router with a wired connection you can use its wifi as hotspot, connect the laptop to Tailscale, connect the Chromecast to the laptop hotspot, and the stream will arrive over the local connection.
3. You can try to install Tailscale on the router, if it runs OpenWRT or something similar, and if it has enough storage space (the Tailscale packages are kinda large). If you announce the subnet from the router then it will work as intended and all the devices on the local LAN will be able to "see" the remote laptop.
4. If you're using nginx as reverse proxy with TLS certificates for the remote Jellyfin you can try this: expose it to the internet without Tailscale and use nginx-ip-whitelister to temporarily allow access from your local LAN's public IP. All devices in the local LAN will see the remote Jellyfin. But read the warnings on the project page, it's not as secure as a VPN. And of course "expose jellyfin to the internet" is not as simple as it sounds (you need a domain, you need to get Let's Encrypt certs, a public IP, a port forward etc.)