Sorry, very late reply here but I 100% agree with you, @f-f , regarding the registry pipeline. Given that pursuit is basically a patched version of compiler generated docs, and given that the registry pipeline compiles everything, it stands to reason that pursuit can be a byproduct of the pipeline.
The tricky thing is doc search. I definitely understand your aversion to the Algolia route because it adds an extra dependency. What I like about Algolia is that it outsources the dynamic bit of Pursuit to a third party, which would allow pursuit to be 100% static modulo the small code snippet one has to insert into an Algolia powered site, which is no more onerous than a Google Analytics snippet.
Another way to say it is that, if we’re looking for the minimal set of of transformations that would need to be done to turn compiler-generated docs into pursuit, Algolia or a similar service may be part of that minimal setup.