Script not running on startup for GPO

Forget about the Startup Script in "Computer Configuration" because the script you set there is executed by the System even before anyone is logged in.

This is typically not interactive (you can add scripts to cleanup certain directories, or configure something on the computer, but you can't start an end-user application here, the app will not show but may be running in the background).

Well, you configured the Logon Script in "User Configuration" too, that's better because the scripts here are executed in the user's context, however, since Windows 8.1 the Logon Scripts don't run for five minutes after a user logs on. This is by design to speed up the loading of the user's desktop. However you can change that:

If you want the logon scripts to run at user logon without any delay, you should configure the Configure Logon Script Delay setting to Disabled in the Computer Configuration\Administrative Templates\System\Group Policy location.

But, still, this is not the preferred way of achieving what you want, as Microsoft says:

This mechanism allows administrators to perform various maintenance and management tasks on client computers, including (but not limited to) collecting diagnostic information, invoking security scans, cleaning or resetting system state, and installing tools.

you are out of scope because you don't want to "perform various maintenance and mangement tasks" with your use case.

I recommend you to deploy a Run Registry Key instead. You can still deploy that with Group Policy:

If you want to deploy this on a Computer (so that it applies to everyone logging in on the computer), then, in your GPO, go to Computer Configuration -> Preferences (not "Policies") -> Windows Settings -> Registry -> New Registry Item, and create a new REG_SZ value under the path shown in the documentation (HKEY_LOCAL_MACHINE\Software\Microsoft\Windows\CurrentVersion\Run), give it a meaningful Value name, and set the Value data to be the path of the executable on the computer.

If you want to deploy this setting on a per-user basis, then use the same method but on User Configuration instead of Computer Configuration (and HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Run instead of HKLM).

Don't forget to link the Group Policy on an O.U containing your Users if you are using User Configuration, or on a Computer O.U if you are using Computer Configuration (same for the Delegation rights, add users if you are working with User Configuration or Computers if you are working with Computer Configuration)