Op-ed: No. You don't need Geoserver. The main benefit of using Geoserver is operating a service that provides many different kinds of layers to different users, and being able to manage them through a web interface. Every time I have deployed Geoserver I have regretted it.

Geoserver does not really do much directly: it's just a kind of web glue for various kinds of backends. So for tile generation, you're better off using Martin (https://maplibre.org/martin/martin-cp.html) or similar.