Hi Peter,
if process is able to write in left top corner system name in unattended run, then is not a RDP issue.
From my experience, while SAP is opened, from time to time, login window fields have different path than standard i.e. conn[1] instead of conn[0]. Following scenario would suit to your error ""Specified SAP Gui component not found"".
Try to kill sap, wait 1 sec and try again, 3 attempts should manage that ;)
Rgrds
L