File formats evolve at a glacial pace and suffer from immense historical baggage. No one benefits from pretending that file formats are as accessible, agile or powerful as proper web APIs.
You are everything that is wrong with modern computing. A web API is always inferior to what you can do natively.
To say nothing of the fact that "APIs", Web or otherwise, aren't actually a means of storing data; at least in the context of things like "pictures" and "music", we obviously can never have "turtles all the way down."
I italicized proper there because my real point is that you're simply defining yourself the win with the word "proper" and not comparing like to like. You can win any comparison by accounting for only the good of one proposition and only the bad of the other, but that's not a valid argument. If, for instance, we ask our web API to have some sort of assurance that it'll be there next year... hell, next week... suddenly "proper web API" turns out to resolve to absolutely no concrete noun at all. (How "proper" would you have called the Twitter API last week? You know, back just before Twitter started shutting huge swathes of third party apps out of it?)
But that's just one definition. The truth is you need the right tool for the job.
It is all, in the end, just bits. The bits don't have color [1], and they don't acquire color by being in a web API instead of a file.