Afaik the Stadia controller talks to the service directly and not to the chromecast. Any other normal setup will be more laggy than the Stadia Controller + Chromecast setup.
There's no bluetooth with the Stadia gamepad, so it's either wired via USBC or WiFi direct to Stadia servers.
You can bring your own bluetooth gamepad if ya want, though.
Afaik the Stadia controller talks to the service directly and not to the chromecast. Any other normal setup will be more laggy than the Stadia Controller + Chromecast setup.