I want to own the database that my apps use
orndorffgrant.com
orndorffgrant.com
I guess you are right about APIs, they eventually won't provide what you need. I'd love to see Solid starting to be used for the apps I use, but honestly I don't feel like it will happen. I don't understand much about Urbit, but from what I read that sounds like another world.
That said, I think there's some startups doing some work with regards to data privacy and security (and it seems like it caught a lot of attention!). As for getting access to the data itself, I think a sort of "open-source" and "reverse engineering" approach would help here, meaning that if you have a proxy between you and the app's server, and supposing you can add plugins to this proxy, you could hijack the data, make sense of it and store somewhere else for your own usage.
Obviously, any external processing of the data would have to be granted by the app itself, but I have a feeling this is what their APIs would contribute with.
I'd be more than happy to talk more about this and even spend some time implementing a PoC, let me know if there's anyone else interested here and we can exchange ideas and/or code/docs/etc.