Uber App Stopped Working, Won't Reinstall

I have an iPhone 5S, fully up to date (version 8.1.2). Recently my Uber app told me to go ahead and upgrade it if I wanted to keep using it. I clicked the "upgrade" button, but nothing happen. Tried again, nothing happened. The dialogue box went away but it didn't upgrade at all.

I've contacted Uber about this first, obviously, but all they've done is exacerbate the problem. They had me uninstall the app, hard reset, and then emailed me the link to reinstall the app. Every time I visit the link and click the "yes, download this app" dialogue option that pops up, the same thing happens: dialogue box disappears, then nothing. No app.

So now I have zero Uber app on my phone and Uber's trying to tell me I just have to lease an iPhone from them, which is absurd since I own an iPhone. I'm not sure if anyone on this end of things can help, but since Uber's tech support was so unhelpful I figured I'd give it a shot.