Implement `file_system` for the web using IndexedDB to store the files

## 🏗 Enhancement Proposal & Pitch

Currently, `file_system`'s web implementation uses `MemoryFileSystem`. Migrating it to use IndexedDB would enable true web caching and make the implementation native.

### Platforms affected

🌐 Web

Start by locating the web implementation of `file_system` and its current `MemoryFileSystem` usage, then review how the platform-specific code is organized. Define the IndexedDB persistence behavior and verify web caching with tests; the issue names no files or tests, so the scope needs clarification first.

Written by the indexing model from the issue text.