This is just server side search with more steps, since the index needs to be selectively split and returned, and then search results page generated from the index.

Sorry, I don't mean to come across as disagreeable. You're right, nothing is perfect, and this is obviously a workable and usable solution. My issue is if we analyse it beyond "it looks like it works", it starts to look like a slightly worse solution than what we already had.

Nothing wrong with moving backwards in some direction, as long as we can clearly point to some benefit we're getting elsewhere. My issues with SSGs is most of the benefits they offer end up undermined if you want any interactivity. This is a good example of that, as you end up compromising on build time and page load time compared to an SSR search solution.