I don't think the controller supports (or is designed to support) such captive-portal networks. In fact, neither does the Chromecast, as far as I can tell. Were you able to get the Chromecast Ultra connected to said public Wi-Fi network?
Hi, the point here is that Stadia is supposed to be portable. Isn't it? So I was at the coffee shop and I wanted to play Stadia in my Pixel phone but it doesn't work connected to USB, it doesn't work via Bluetooth... only wifi... IT'S RIDICULOUS!!
I don't think that is actually the point. If it were, then the Stadia app would support playing over 4G rather than just WiFi.
And right now, the controller only works on WiFi when you're playing on a Chromecast. You'll need to use a USB cable with the controller if you want to game on Chrome or your mobile phone.