The reason that most extensions exist is that there is (or was) no way of implementing things that people want with standard C. Inline assembly is one example. All modern C compilers support it, but GCC and Microsoft's compilers use different syntax (most other compilers implement one or the other, sometimes both). Without it, you require that every time you want to use even a single instruction of platform-specific assembly code, you must write an entire function and call it.
Atomics were another big reason for extensions. Prior to C11, if you wanted atomic operations, you needed either assembly or non-standard compiler intrinsics. Efficient vector support is another one.
Please correct me if I'm wrong because I may not have imagined this system properly. I was thinking the idea was that you encrypt each file with a single unique key, and then to use a public-key encryption scheme to encrypt that key. You can then send the encrypted file and the encrypted key to another user, knowing that it will need that users private key to decrypt.
Every time you upload a file, you generate a random symmetric key. You encrypt the file with this key and the key with your public key. If you want to download the file, you get the file and the encrypted key and then you decrypt the key with your private key and then decrypt the file. When you create the account, you upload your public key.
When you want to share a file with everyone, with no access control, you download the encrypted key, decrypt it, and provide it to the server. The server can then decrypt the file.
When you want to share a file with a limited set of users, you download each of their public keys (which you can cache in the client) and the encrypted symmetric key, decrypt the key, and then encrypt it once for each user. They will then only be able to access it with their client.
I'm not sure who you're 'we' as in 'internet community' is. We do have standards and off-the-shelf libraries for everything required to implement this and others have done so in the past (one of my colleagues during her PhD did back around 2006, to give one example, others have implemented more complex and flexible schemes more recently). Note that this is the simple textbook scheme for doing this kind of system. It's been implemented before and doubtless will be again. If you check the research literature then you'll find more interesting schemes.
The only problem is if you want to be able to access it from the browser, without some kind of plugin (Google actually does compile OpenSSL with Emscripten to do ASN.1 parsing, but I wouldn't recommend using it for encryption).
then you may as well just give the server the AES key and ask it to decrypt the file
But in that model, if "the server" has the key, wouldn't Dropbox have the key? I thought that was the whole thing people were freaking out about.
No, you'd have the key. If you wanted to share the file publicly, then there's no point in keeping it encrypted, so you'd provide the server with the key and it would decrypt, saving you the cost of downloading and reencrypting.
I understand what you (and the AC) are saying about storing an encrypted key on the server, and then re-encrypting the key for each new user you'd want to share with. That's a clever arrangement and I admit that I hadn't thought of it, but it still seems like it has the potential to create more complexity than most people want to deal with. It still means you need to manage various encryption keys, and we (Internet culture) seem intent on not developing a coherent system for managing encryption keys.
The client just needs one key, the RSA (or equivalent) public key. You'd need to copy this between devices, but it's relatively small (under 1KB). It's small enough to fit in a version 40 QR code quite easily, so you could set up mobile devices by displaying the QR code on your laptop screen and point the mobile device's camera at it, if you don't have any sensible way of transferring files between devices. The client then has to download the file and the associated key, decrypt the key with the locally-stored key, and then decrypt the file, but that's not something that's exposed to the user.