#3 was not necessary. Even if it was manually installed before, the assignment still occurs, and the installation becomes managed (i.e. controlled by the group policy). It then attempts to reinstall it, which results in simply discovering that it already is installed.
If you uninstall managed application, it still remains assigned to the machine, and you will still be seeing it in add/remove programs. However, it becomes advertised then. If it has advertising information, such as advertised shortcut(s), file extensions, or OLE, then activating any of those will result in install-on-demand, and it will be reinstalled. But winlogon won't be making another attempt, even if you initiate "redeploy" in the group policy.
You can force re-installation either by activating an advertised piece in it (any of the above-mentioned), or by running command
msiexec.exe /q /i {product-code}
You can find product code by looking in the package itself, or from the registry at the workstation where it's installed, under HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall - find the product and look at the UninstallString.
You can run the command using psexec, or from logon script.
