Let’s ask for an official digital DRM-free rerelease.
https://www.gog.com/wishlist/games/gadget_trial

How can I play it in Japanese? The game comes with the English patch already applied.
I assume I’m meant to use the English patch uninstaller, but when I try to do so it gives me an error stating that It couldn’t find the uninstall data.

Yes, the game in rar is laid out ready to play english patched version, but original japanese files are still included.

You need to rename (and replace) either Backup/GADGETT.BAK or GADGETT.old to GADGETT.exe (both work similar, I don’t know the difference, but the files are different, and GADGETT.old is identical to GADGETT.exe from original CD image), and all Backup/*.BAK to Data/*.PAK.

You need to run exe in japanese locale, for example with Locale-Emulator, or you’ll see garbage text (in dialogs and window tab title for example). Some of the text looks okay even then (game exit dialog), and some is still english (same as in original version from CD image, I checked, its torrent is now deleted on Nyaa, but you don’t need it, the game will be the same, only with original installer). Still, with locale emulation, some text (dialogs, but not dialog history) will look garbage, so further tinkering is needeed, maybe install some fonts, or full system locale change, or a complete japanese OS installation, which I won’t. Write back your results if you try.