Can the original Gameboy be played with an AC/DC adapter?

I recently found a discussion on Twitter where a user claims to have bought a power adapter for her Gameboy, and many people criticizing her because the Gameboy used AA batteries.

Now, I'm too young to have used the console (my first console was a Gameboy Advance), so I have this question.

Is it possible to use a Gameboy with an AC/DC adapter? If so, was the adapter used to charge batteries or only to power the console itself?

As Wikipedia clearly states, the original Gameboy can be used with an AC/DC adapter:

The Game Boy also contains optional input and/or output connectors. On the left side of the system is an external 3.5mm × 1.35mm DC power supply jack that allows users to use an external rechargeable battery pack or AC adapter (sold separately) instead of four AA batteries.

Here's an image of the official power supply sold for the system:

image source

Yes, the Gameboy has a DC jack:

However, it is one with a built-in switch to switch between the batteries and the DC input. This switch isn't perfect and the Gameboy has very small (nonexistent?) supply buffer caps, so plugging/unplugging the adapter during a game usually crashes or resets the game.

It is not used to recharge the batteries. (The Gameboy can be powered from non-rechargeable batteries.)