For your MacBook Air 13-inch (M1) running macOS 27.0, I would work through this using some of Apple’s own troubleshooting guidance rather than assume that Golden Gate is the cause.
When an app refuses to quit, first try Command-Q or the app’s Quit command. If that does nothing, press Option-Command-Esc and choose Force Quit.
If the app still will not close, Apple recommends restarting the Mac. Apple Support — How to force an app to quit on Mac
There is one macOS 27 feature that is particularly relevant to what you described.
Apple says that some apps can continue running in the background after you quit them, and macOS 27 indicates this in the Dock.
Background activity can also affect Mac performance.
You can review this under System Settings >> General >> Login Items & Extensions >> Background App Activity.
This is different from an app that has actually become unresponsive. Apple Support — If an app continues running in the background after you quit it
Since you are also seeing frequent system-wide lag, open Activity Monitor while the slowdown is happening.
Apple provides it to examine CPU, memory, energy, disk, and network activity.
Also check System Settings >> General >> Storage, since Apple identifies insufficient available storage as one possible reason a Mac may run slowly. Apple Activity Monitor User Guide Apple Support — If your Mac runs slowly
If the problem continues, try Safe Mode. On your M1 Mac, shut it down, Do Not use Restart, hold the power button until the startup options appear, select your startup disk, hold Shift, and choose Continue in Safe Mode. Apple says Safe Mode can help determine whether software that loads during startup is involved.
It is also worth reviewing System Settings >> General >> Login Items & Extensions to see what is allowed to open at login or run in the background. Apple Support — Open items automatically when you log in
If the problem persists, Apple provides Disk Utility for checking and repairing the startup disk when appropriate. Apple Support — If your Mac runs slowly
When an application becomes unresponsive, Activity Monitor can create a Spindump report, and System Diagnostics can collect additional information that may be useful to Apple Support. Apple Activity Monitor — Run system diagnostics
If the problems remain after these software checks, Apple Diagnostics can test the Mac for possible hardware problems. Apple Support — Use Apple Diagnostics to test your Mac
I would handle the Siri problem separately by checking the internet connection, Siri settings, and microphone operation rather than assuming Siri is causing the overall slowdown. Apple Support — If Siri on Mac doesn’t work as expected
Finally, check System Settings >> General >> Software Update and make sure your apps are up to date.
Apple recommends checking for updates when an app repeatedly stops responding. Apple Support — If an app on your Mac stops responding, closes unexpectedly or won’t open
This gives you a practical Apple-supported path through the issues you described—apps not quitting, apps becoming stuck, persistent lag, background activity in macOS 27, storage and disk checks, startup software, Siri, and possible hardware problems—without assuming in advance what is causing them.