(Mobile Safari) can upload pictures, but that is it. Mobile Safari needs a way to upload and download any type of file into iCloud Drive (a mirrored from the Mac downloads folder). - Bradley Chambers
I currently use Readdle Documents to download files Safari can't handle. It's built in browser can handle any type, and the app itself can easily push those files do Dropbox, iCloud Drive or any other cloud based host.
But Safari kinda has the same feature already. You can download a file and use the open in... menu to save or upload the file to a different app or platform. I often use Dropbox as a destination so I can save the file and have it available on my Mac.
The downside is when you do this, you have to first download the file, and then upload it to the other service. Twice the data for one file.
A nicer solution would be a scenario where when I long press a download link,I get a list of iCloud enabled devices I own, and select a device which would download the link over a better connection and save it to a shared location. A sort of hand-off for downloads if you may.
Which brings up another question: I don't understand how iCloud Drive handles caching. What if I download a large file on my Mac (a movie for example, or an Xcode beta) and save it to the iCloud Downloads folder. Is that file synced to every device I own? Also a 16GB device?
Either way, an iCloud downloads accessible from within Safari solves the download issue. But there's also uploading. As Bradley writes in his article: There's currently no way to access other apps from within Safari. And as far as I know the new extensions won't change that. What could work is integrating the document picker in Safari. When you click on Upload on any webpage, instead of only providing your Photo Library, you could then also access any other file system installed on your iPhone. iCloud, Dropbox, OneDrive.
I think this is yet again an example of Apple being on the right track but not there yet. They massively improved with iOS 7, added extensions and iCloud Drive in iOS 8. So maybe iOS9 wraps it all nicely together?