Replace manually maintained types in src/lib/types.ts with auto-generated
types from FastAPI's OpenAPI schema using @hey-api/openapi-ts. The bridge
file re-exports generated types with renames, Require<> augmentations for
fields that are optional in the schema but always present in responses, and
manually added relationship fields excluded from OpenAPI.
- Add openapi-ts.config.ts and generate:api npm script
- Generate types into src/lib/api/generated/types.gen.ts
- Rewrite src/lib/types.ts as bridge with re-exports and augmentations
- Fix null vs undefined mismatches in consumer components
- Remove unused manual type definitions from api.ts
- Update AGENTS.md docs with type generation workflow
