i don't have any experience with geese. but it seems i see more about people trying to rid themselves of geese then trying to keep them around sometimes.

but they are supposed to be best for weeding and mowing.....
I think either ducks or chickens could work for you. I had ducks for just a while and found them to be too messy for my situation at the time. they found and mussed up all the water sources for the other animals. but you may not have to deal with that problem. and I plan to get ducks someday again when I can arrange things a bit differently. I want them mostly for their superior slug eating capabilities.
if you are not wanting to breed then there is no need for a rooster if you have chickens and that will avoid the crowing if it bothers you.
I can attest to the fact also that no young plants can be considered safe from either chickens or ducks. the chickens scratch them up after bugs (they love any loosened dirt) and my ducks flat out ate my lettuce and cabbage etc.....
with careful planning and fencing those problem can be mitigated.
some people specifically utilize geese to weed. they are sold in the catalogs as weeder geese. I dont' think they are a particular breed. they are whatever is leftover that they didn't sell.
it seems they each have their pros and cons.
I have chickens. they are easiest for me right now.