As you've discovered, Chromecast doesn't support 2-step (also called "captive portal") logins like that where you connect and then have to go to a web-page to finish authorization.
If you do a google search for "use Chromecast in hotels" you'll find how people have worked around this using phone hotspots and (my preference) travel routers.
Keep in mind that neither solution is officially supported and most shared networks like that are grossly overloaded so streaming may not be a smooth experience.
Also, you don't really WANT to get connected to a big shared network like that. If you did, your Chromecast would be visible to everyone else on the network and they could cast to it at any time. Using a travel router makes a private network of your own and then nobody else will be able to see your Chromecast or cast to it.
Another huge benefit of using a travel router is that the network it makes is known to all your devices. Once I set mine up, I don't have to deal with logins for anything else I happen to have with me; my phone, tablet, Chromebook, e-readers, laptop, Chromecast devices, etc. They all know the router's network and login automatically once they see it.
