Implement "Browse Product Catalog" (UC-010)

Problem Description

An online shop's engineering team has already run /entity-model and /drizzle-migration for their NestJS + Drizzle + Next.js project. The product table already exists in the Drizzle schema (apps/api/src/database/schema.ts), the entity model is at docs/entity_model.md, and the use case is specified at docs/use_cases/UC-010-browse-product-catalog.md. Nobody has written any backend or frontend code for this use case yet.

The repository is an npm workspace with two applications. The backend at apps/api is NestJS 11 on ESM/NodeNext with Drizzle ORM over PostgreSQL; its existing categories feature shows the conventions this team follows. The frontend at apps/web is Next.js on the App Router, with an existing but empty products route at apps/web/src/app/products/page.tsx. The frontend reaches the API through the /api/* rewrite configured in apps/web/next.config.ts.

Output Specification

Implement UC-010 end to end:

1. Backend (apps/api/src/...): a products feature that reads the existing product table, applies the "available only, optional category filter" logic from BR-010, and exposes the result as JSON through a REST endpoint returning a response DTO — never the raw database row.
2. Frontend (apps/web/src/app/products/page.tsx): fill in the empty page so it fetches from the new endpoint and renders the product list with a category filter control, matching alternative flow A1.

Do not create any test files — that is out of scope for this task.