How would you share data between two apps if both developers didn't support that?
On Android, data sharing is fully handled by the OS, not unlike the copy/paste buffer in most desktop OSes. This means the list of applications with which one can share data is consistent both in terms of content and capability.
Something I've observed to be true is that iOS applications seem to be specifically coded to share data with other iOS applications. A lot of things can share data to Dropbox, but fewer seem to be able to share the same data to Google Drive or Onedrive. Data sharing seems to be a one-way street where the application developer has to support whatever hooks were provided for the target app. At the very least, the list of supported applications for sharing does not appear consistent from app to app, even within the context of a particular data type. I suspect this is in large part due to the iOS security model, but I take issue with that for other reasons anyway.
Everything you mention is fine but I'm not sure there's some killer user story or use case that justifies it in light of the security issues. I don't think any 3rd party app developer should be able to see any of your file system ever, not on your phone. It's just too dangerous, the thing is always on the network, it knows where you live and you can't unplug it.
"I can't think of a reason someone would want to to it, so it must be a bad idea."
There's a 128GB iPad sitting in my office. I have no particular use for a 128GB iPad, but it's still 128GB of flash storage that I could potentially use for something-or-other (yes, I am aware that I can get 128GB flash drives for under $50 but that's what a 128GB iPad is worth to me). Putting that aside, it's storage. On the iOS device, I have to associate everything with a particular application. I can't even use the stupid thing to transfer inert data (that I already had to add through iTunes since the device can't meaningfully interact with SMB, FTP or NFS) that for one reason or other doesn't match up with file size limits on my cloud storage provider's service.
Likewise, I don't have any control over arrangement of data under iOS. I have to accept whatever the device does and like it. That sometimes means making multiple copies of the same file (on a device that's specifically sold on the basis of its storage limitations) for different apps in cases where those two apps can't share data. It also means potentially jumbling a lot of data together that I don't really want to have view that way. Should I really be forced to reorganize my data to conform to the limitations of the device?
Whether or not applications are granted the ability to access a filesystem, the system owner should be able to do that even if it's just an infrequently used option.
Honestly I just spent about 30 minutes trying to find a website where I could even try to download an MP3
Ahem. This is a thing that people do. This is a thing people do all the goddamned time. Yes, you can get an app on iOS that can sandbox those particular downloaded MP3s on internal storage, but look at how ridiculous the workflow is to move those files out of that sandbox and in to the default music app so you can add them to your normal playlists.
Even speaking of podcasts, haven't you ever been browsing on your iThing and wanted to snag a one-off episode of something? "Oh, I want to download the rest of that episode of Fresh Aire that I heard 10 minutes of in the car. Should I open a podcasting app and then the Fresh Aire feed so I can find that one episode that was a rerun originally recorded in 2007 and therefore buried in the feed or should I just search for it from the web?"
One app to play all your music is 1990s thinking; modern apps are meant to brand content and service experiences, instead of them all launching the "native music player" they all call the same native sound API. The mechanics of how the media moves across the internet or across the filesystem is invisible to the user.
My personal tastes are sufficiently niche (contemporary classical music) that I am not well served by music-as-a-service. Nor can one assume that mobile devices have ready access to a high-speed data connection. Many people have bandwidth caps or spend the majority of their time in places where data services are too slow, unreliable or unavailable. That being said, on other mobile platforms I am familiar with, adding music or playlists to a device makes that music or playlists accessible to all audio players on the device. Amazon or Play Music that is copied locally is exposed to each app in turn as well as other player apps that might be installed such as Doubletwist or WinAmp.
As far as I can tell, the only ways to add music to whatever the hell the Apple players are called is through either the iTunes desktop application or from the iTunes store. It's possible to get a third party music application, but that application will be sandboxed so that music used with it is simply not available to the rest of the device. Nothing outside can share music to the native apps, not even convenient tools like Safari and Dropbox. You HAVE to plug the device in to a PC, export and import.
These are more than legitimate issues. I understand that you see them as defensible and that you've adapted whatever personal mobile device workflow you have to Apple's anointed path, but there really are common usage scenarios where iOS is somewhere between moderately unfriendly and actively hostile to user needs.