You don't need to push every time you spawn, only when the code is changed. Typically, you would use a CI/CD action to push the image (this repo has a simple example: https://github.com/drifting-in-space/jamsocket-jupyter-noteb...)

A typical FaaS is stateless between invocations. We have an explicit "spawn" step because each spawn produces a new server process with its own state and own DNS hostname.