The premise contains a misunderstanding: **Action Replay devices are not made *by* Nintendo for the DSi.** They are **third-party cheat cartridges** manufactured by companies like Datel. Here's why they exist for the DSi:
The DSi is **backward compatible** with almost all physical Nintendo DS cartridges. Action Replay DS/DSi units are **DS-mode flash carts** that plug into the DS cartridge slot. They patch DS game code in real-time on the DSi (running in DS compatibility mode).
Datel released updated carts (e.g., Action Replay DSi/3DS) with firmware that bypassed Nintendo's blocks. They still only cheat **physical DS cartridges**. They do not work on DSiWare, 3DS games, or the DSi menu.
Action Replay exists for DSi because **millions of DS cartridges are playable on DSi hardware**, and cheat manufacturers want to sell to that audience. It is not an official Nintendo product, and its functionality is limited to DS-mode cartridge games.