Why are those things mutually exclusive? You can have an API and then have a different app that uses that API with server-side rendering. I.e. instead of a client SPA you have a server app using the API.
Sure they're both viable options, but running 2 server-side apps that could be 1 server-side app and a folder full of HTML/CSS/JS served over CDNs certainly has some trade-offs. e.g. in terms of security, I'd prefer to have a smaller surface area server-side.
Then you're comparing server-side apps to client-side ones. And imo the attack surface of managing a SSR webapp is fairly minimal - especially when you're already responsible for providing an API to the public. My point was more that API-first doesn't prevent you from doing server-side rendering and there are benefits to doing so.
