[base-controller] Make `name` a static property

The `BaseController` API has `name` as an instance property. This is always set statically, so we could make it a static property instead.

This would make it easier to generate a complete list of controller names in the client repositories. Currently we need to wait until after the controllers are constructed before getting a list of the `name` properties, which is an inconvenient restriction.

Start by locating the BaseController implementation and references to its name property in the monorepo. Review how controllers are constructed and how client repositories obtain their names. Done means name is available statically before controller construction, with affected tests and usages passing.

Written by the indexing model from the issue text.