If the data "belongs" to the server, why not send the query to the server and run it there?

We could, but if the data size is not that huge, sending it once to the client and then letting the client perform the queries can be desirable. The tool works without internet access, the latent is much better, all results are coherent etc

If the data "belongs" on the client, why have it in database form, particularly a "data-lake" structured db, at all?

Just because it fits in memory doesn't mean that the shape of the data does not matter. A data structure optimised for whatever query/analysis you want to perform has a significant impact on how fast and efficiently you can perform those operations