I agree, geese are not really 'protectors' like you are thinking. They are good alarms usually, they will let you know if something is not right. Not so much as guineas would, but they usually sound an alarm. If it were me, I'd get a couple of young guineas to live with the chickens, so when you let them out the guineas would stay with them and alert the chickens to danger. Of course, the chickens would have to 'learn' the guineas ways, and there's always the chance that the guineas won't hang with the chickens, but if you get young ones (2-3 months old at most), they should see the chickens as their 'flock' or 'mamas' and stick around.
Get a goose because geese are cool, not because you think they'll protect your birds. If anything, my geese will run away if they think they are in danger, they won't hardly protect themselves.