The debug overlay appears inside menus even when the HUD is hidden

Affected versions: 1.21.9+

In the Copper Age update for Minecraft 1.21.9 the Debug Screen (also known as the F3 screen) was entirely revamped, containing less "clutter" and being more configurable. Since that update, however, the F3 has experienced a slew of issues, this is another one of them.

Steps to reproduce the issue:

1. Boot up any Minecraft version lower than 1.21.9 or prior to 25w31a, where the F3 changes were implemented.

2. Enable the Debug screen by pressing F3 and hide HUD by pressing F1.

3. Open any container or pause the game.

Observed Behaviour (Expected)
The F3 screen is not visible when being inside of a different menu whilst having F1 enabled.

1. Boot up any Minecraft version after 1.21.9 or after 25w31a.

2. Enable the Debug screen by pressing F3 and hide HUD by pressing F1.

3. Open any container or pause the game.

Observed Behaviour (Actual)
The F3 screen is visible when being inside of a different menu despite having F1 enabled.

For a better example, refer to the attached videos.