Although iTunes was the original method for copying files to/from iOS devices, its not the only way and not the only issue.
The real issue is that iOS doesn't expose the filesystem, and apps have their own sandboxes for files. So when you copy files to the device, where you put them determines which single app is going to be able to open them. There may be some slight exceptions to this, but thats mostly how I understand it.
Cloud stuff is the main way to deal with this issue with as little pain as possible, whether it be iCloud, dropbox, some other service or your own NAS on your local network. But you need to make sure whatever app you want to use to open that sort of file is compatible with whatever cloud/storage option you are using.