Splitting it up sacrifices readability. After splitting it up, the code is scattered over a bunch of methods, while now you can just look and follow the lines one by one to understand how it works. I don't see any cases where we would reuse part of this method instead of all of it.

Testability is a legitimate concern. However, I think the current coverage is fine. The remote tests test the implementation of this method, while for unit tests we stub out the entire method. Potentially we could split out the response handling, so we can unit test that separately, but it is so simple that the remote test coverage of that is fine IMHO.