[Sync Input] Synced screens don't handle Quick Command replacement properly

Description:

When using Sync Input in combination with Quick Command, synced screens do not replicate the behavior correctly compared to typing directly in the active screen.

Steps to Reproduce:

Enable Sync Input across multiple screens/tabs.

In one screen, type a few characters to trigger the Quick Command popup.

Press Enter to select a Quick Command.

Observe the behavior on the other synced screens.

Expected Behavior:
The synced screens should behave the same as the screen where the input was made — i.e., the typed characters should be replaced by the selected Quick Command.

Actual Behavior:
On the synced screens, the typed characters are not replaced. Instead, the result is the originally typed characters plus the selected Quick Command appended after them.

Example:

Type abc → Quick Command popup appears

Press Enter to select a Quick Command (e.g., hello)

Active screen: hello (characters replaced ✅)

Synced screens: abchello (characters not replaced ❌)

Number of synced screens: [e.g., 1+]

Additional Context:
This issue makes Sync Input unreliable when using Quick Command, since synced screens end up with inconsistent and incorrect input.

Start by reproducing the issue with Sync Input enabled across multiple screens, then trace the Sync Input and Quick Command handling paths. Done means selecting a Quick Command replaces the triggered characters on every synced screen, matching the active screen.

Written by the indexing model from the issue text.