Issue: Steam Flatpak Game Backups Appear Successful, But Folders Are Empty

### Description

I'm running **Fedora 42 with GNOME** and using the **Steam client installed via Flatpak from Flathub**. The client generally works well, including Proton for non-native games. However, I'm unable to create proper backups of my games for later restoration.

### Steps to Reproduce

1. Launch the Steam client (Flatpak version).
2. Navigate to the game backup and restore feature.
3. Select one or more games for backup.
4. Choose a destination folder for the backup (e.g., an external drive or a specific folder within my user directory like `~/GameBackups`).
5. Proceed with the backup process.

### Expected Behavior

The Steam client should successfully create a complete backup of the selected game(s) in the specified destination folder, with all game files present.

### Actual Behavior

The Steam client reports that the backup was **"completed successfully."** It also creates the intended game folder structure within the chosen destination. However, upon inspecting these newly created game folders, they are **completely empty**. No game files are copied, despite Steam indicating a successful backup.

This issue doesn't occur when using the **native RPM version of Steam installed via RPMFusion**. This leads me to believe it's a specific bug related to the Flatpak packaging of Steam, possibly concerning permissions or file system access within the Flatpak sandbox. As I predominantly use Flatpak applications, I'd appreciate this issue being investigated.

### Additional Information

* **Operating System:** Fedora 42 (GNOME)
* **Steam Client Version:** 1747701111
* **Flatpak Version:** 1.16.1
* **Source of Steam Flatpak:** Flathub


Start by reproducing the backup flow with the Flathub Steam client on Fedora 42, using both a home-directory folder and an external drive. Compare the Flatpak behavior with the RPM version and inspect the app's filesystem permissions and runtime logs. Done means selected games produce non-empty, restorable backup folders and the cause is documented.

Written by the indexing model from the issue text.