Synergy requires the 'Accessibility' permission for:
$HOMEBREW_PREFIX/opt/synergy-core/Synergy.app
You can grant this permission by navigating to:
System Preferences -> Security & Privacy -> Privacy -> Accessibility
If Synergy still doesn't work, try clearing the 'Accessibility' list:
sudo tccutil reset Accessibility
You can then grant the 'Accessibility' permission again.
You may need to clear this list each time you upgrade synergy-core.