The computation for rendering (in every case I've seen, and I have to speculate in 80% of cases ever) is so trivial compared to the actual retrieval of the data to be rendered.
More seriously, though, it's nice to be able to just build without thinking too hard about if you're getting your abstractions perfect. To me, this is the main advantage of SSR - moving fast doesn't leave behind a wake of idiosyncratic APIs that need to be (carefully, dangerously) cleaned up later.
You still need client-server communication, so you still have an API, it's just an ad hoc API that speaks HTML and form data instead of JSON. And because you didn't think of it as an API while you were building it, it actually tends to be harder to clean up later, not easier.
Just pass a header for request content type json. Then the server returns data in json format as opposed to html.
Aside from all other implications, letting each client render the same stuff is a massive waste of energy and compute.
Either way, the measurement of Joules/page is likely to be such an astronomically small number compared to the constant cost of simply having a server at all IMO.
I've done this multiple times and it's fine. It's really not an issue for delivering projects. I'm not saying I'd do this every time - if the API was identical for both my app/website and any other clients I'd very possibly abandon the SSR approach for the ease of a frontend API that everyone can use identically, but that's a specific project requirement not a general statement on SSR and API development.