This just runs Postgres as a process, right?

Not OP, but I think it does run Postgres as a process. However, IMHO the general use case for SQL is for external actors (humans, machines) to get access to the underlying data in a structured way. So I see a benefit for a true in-process embedding of Postgres if the process exposed a Postgres TCP/IP port 5432, etc. (Hook your software up to a query tool, a reporting interface, etc.)

Beyond that, why care whether the "embedding" involves a spawned process? It still works great for integration tests which I suspect is the main use case, and for specialized data analysis software where a spawned process is no big deal.

Yes, but it embeds it in your executable so it is transparent to the end user.

UPDATE: Actually, I see it is downloaded (and I think cached?). I can't recall if you can embed as an option or not.