The value of having your own data is, precisely, that you have your own data; nothing more and no less.
Furthermore, and although I'm no expert in Solid's and other implementations, it's quite likely that you, or other technologists, can more easily help you achieve the interoperability you so desire if your data is not locked away behind thousand APIs and rate-limits; after all you do have your data in your pod. Think of Instagram, what if all your data was stored locally? Even if it was stored in a really messy JSON schema or whatever you can think of; it's still trivial to massage that data to import it into Wikimedia, or iterate through it and send it in batches to your email, or remove it, etc.
I'm not sure how solid plans to keep applications from stealthily generating derivatives of the data they manipulate. I can imagine some sort of separation would have to happen at the browser level where the "app" provides a scaffold and the browser replaces in your data.