Unfortunately for us iDevice users, invisible background tasks like that are a no-go. Apple simply won't let you ship an app that sits around and invisibly logs you in like that.
Fortunately, iOS itself is smart enough to notice when your network has a login page like that, and will display that page on-screen automatically so you can click through it. Not as smooth as that Android app (which I'm now jealous of), but at least things don't invisibly fail until you figure out what's going on.
It's likely going to get much, much worse. Try to sign in to Starbucks wifi in Italy & much of continental Europe.
That's the future, sadly - Italy for one requires you to register with your passport or other ID in order to access free wiki. It's an anti terrorism measure. Every hotel you stay at requires you to sign on individually, in a manner connected to you personally, every single time.
Fortunately, iOS itself is smart enough to notice when your network has a login page like that, and will display that page on-screen automatically so you can click through it. Not as smooth as that Android app (which I'm now jealous of), but at least things don't invisibly fail until you figure out what's going on.