Is there an ALLUSERS switch you can add on? It’s likely as Tom mentioned, it’s only adding shortcuts under the account used to install it.
If there isn’t switch to install for all users you could write in your batch the commands to copy to the public desktop and start menu.
Assuming that messenger-setup.exe and the shortcut file are in the same root location as the installation script, something like this should work:
CLS
ECHO Installing School Messenger... Do not close.
ECHO.
CD %~dp0
START "School Messenger" /wait "messenger-setup.exe" /verysilent
SLEEP.exe 10
COPY "School Messenger.lnk" "C:\ProgramData\Microsoft\Windows\Start Menu\Programs\School Messenger.lnk"
COPY "School Messenger.lnk" "C:\Users\Public\Desktop\School Messenger.lnk"
CLS
ECHO Installed School Messenger
ECHO.
SLEEP.exe 5
EXIT
If that doesn’t work, maybe provide the application so others can test it.