Whether you need a spinner or not should be decided by the User Experience (e.g., when the user has to wait for more than 100ms, show a spinner), and not by the location of the data. I am a big fan of local-first apps and enjoy building them myself. However, sometimes your app takes a moment to load. With local-first, you eliminate the network as a source of delays, but there are other factors as well, such as large data sets or complex algorithms.
For example, when you have a project planning software and want to plan 100 work packages with multiple resource combinations in an optimal way, depending on the algorithm, this can take some time. In that case, a spinner or a progress bar is a good thing.