Divide the pkg/server to subdirectories

E.g., divide it in to frontends and backends, and maybe cut a common connection manager for the backend/frontend manager.

Start by mapping the current contents and dependencies of pkg/server, then identify the frontend and backend responsibilities and any shared connection-management code. Confirm the intended package boundaries with maintainers before changing structure; done means the server is split into coherent subdirectories without breaking existing behavior.

Written by the indexing model from the issue text.