Lol, just was debating this at work today.
Company I work at, they want to add a discussion board for the first app. Same questions as yours came up. The frontend devs use react. So they just want an API, backend devs say it's not really built for that (one feature API driven). But backend guys where stating the same thing. What's the next few years going to be? All API?
To be honest, it depends per app but definitely looks like it generally will be API driven backends.
Most large sites like Google (not just search), Facebook, twitter, Yahoo and Yahoo mail, LinkedIn etc are all API backend.
At a very fundamental and basic level vue router just changes the URL and remove/replace components. Basically a spa but allows for requests on component load versus a more traditional all data at once.