"Syncro" button in Settings uninstalls dms-greeter on a fresh install

### Compositor

Niri

### Distribution

Arch Linux

### If Other, please specify

_No response_

### Select your Installation Method

DankInstaller

### Was this your original Installation method?

Yes

### If no, specify

_No response_

### dms doctor -vC

The issue happened one week ago, I didn't save the output, though when I did `dms doctor` that time while encountering the issue there were no error. Only optionnal things missing. I was suprised there were no mention of `dms-greeter` in it.

### Description

About the **Installation Method** I'm not sure. Because it installed itself directly during the installation of Arch Linux live ISO (release 2026.07.01), using `archinstall` and the desktop profile `niri - DankMaterialShell`. I guess it's `DankInstaller`. The greeter is `dms-greeter`.

---

After the installation of the OS I installed my apps and `paru` (AUR). I did my configuration, styling and changed the wallpaper.

Then I wanted to have the same colors and wallpaper on the greeter. So on the settings `Greeter > Status` I clicked the `Syncro` button. After restarting I was 'greeted' with an error saying `/usr/bin/dms-greeter` couldn't be found.

To fix the issue I went in the TTY, installed it manually using `paru -S greetd-dms-greeter-bin`, then restarted again.

### Expected Behavior

The `Syncro` button in the Settings > Greeter > Status shoudn't nuke the binary `/usr/bin/dms-greeter` when not installed manually (or explicitely).

Though I am aware it can be an issue on `archinstall` instead of DMS.

### Steps to Reproduce

1. Install a fresh Arch Linux via `archinstall` and choose the desktop profile `niri - DankMaterialShell`, let `dms-greeter` as selected in the greeter section.
2. Boot/restart into your new OS.
3. Style it (colors/wallpaper).
4. In the settings, "Greeter > Status" click on the button `Syncro`.
5. Reboot the computer, you should see the error `/usr/bin/dms-greeter not found`.

### Error Messages/Logs

After "greeter syncing" and rebooting on a fresh Arch Linux install :
```
/usr/bin/dms-greeter not found
```

### Screenshots/Recordings

N/A

Start at Settings > Greeter > Status and trace what the “Syncro” action does on a fresh Arch Linux installation created with archinstall and the niri - DankMaterialShell profile. Reproduce the flow with dms-greeter selected, then verify that syncing preserves /usr/bin/dms-greeter and that the greeter starts after reboot.

Written by the indexing model from the issue text.