This isn't a web service deficiency by Apple, it's a design choice for simplicity, which can be considered the most suitable form of file management for smartphones/tablets. For instance, instead of backing up bookmarks, my bookmarks are automatically synced to iCloud & across all my OS X & iOS devices. If I add a bookmark on one device, it's immediately available on all other devices. In addition, I see tabs open on all other devices from any device (desktop, tablet & smartphone). I don't need to manage bookmark files. This is a smart use of web services.
If you want to store & manually manage all your files online, there's DropBox et al as well as "hacks" to mount a folder in iCloud & use it like a network drive.
You have just described the behavior of Chrome.
The difference, however, is that I can all of that on my mac, iOS, or Nexus 4. Safari does not exist for Android. If it did I could make it my default browser should I so desire (not so when I use Chrome on iOS). Yet I can use Chrome on iOS. Hmmmm....
I think the person you responded to was talking about freely accessing files from anywhere. This can be done on free cloud services.
Speaking of accessing files... Where are videos stored in iCloud? Certainly not in PhotoStream. They are in iCloud backups if you are lucky enough to have enough room (or pay for more storage) and have waited long enough for a device backup. However, how can you access a video--just one video--that you want to see or retrieve from another device? You can't. You can only do a restore. I don't like that.
On Android photos
and videos are backed up to a Google+ private folder. It's Google's answer to photo stream but in my opinion better since it includes videos. For me it is even better still since it excludes screenshots (hate that photostream backs up screenshots since I use them a lot and do not need them there). In fact Android places screenshots in their own album.
Michael