Here's an example video of what's going on.
LaunchBox 2023-12-12 10-53-29.mp4
Edit: Apologies for the dialogue and child windows not showing, but what I'm doing is right-clicking, edit, then marking the game as installed. Then I "Play" the game, and quit. When I quit, I click the next game cover and go back to GTA, and you can then see that the game has reverted its "Installed" status and offers me to once again install the game. It only reverts after launching and quitting the game. Even though it's offering to Install the game, if I click that Install button, it launches the game just fine.