[Edit: I don't want to get into this argument right now]

the original web architecture has sophisticated client-side caching built into it, all modern browsers support it:

https://developer.mozilla.org/en-US/docs/Web/HTTP/Caching

the difference between constructing a string of JSON and a string of HTML is a round off error when compared with network connection costs and data store accesses, etc.

I'm guessing you haven't measured it, as IME it certainly is true. Try it, you might be surprised.

Also, SSR'd pages can be cached too.

"The server load from a server-side rendered page vs an API call that returns JSON is... very minimal."

That greatly depends how you designed your architecture and if you have just a simple webpage or a sophisticated webapp displaying data in various ways.

My main app is designed to work mainly offline, with bursts of data transfer. All the state is local. Sending html from the server would mean in my case, the server would need to know and keep track of all the user states and data.