I brine. I know why and how to brine, and why not and how not to brine. Those things make a difference.
If your brined birds taste watery and/or overly salted, they were improperly brined. The problem is not the technique, but your technique. If you get the right amount of water and salt into the bird, it will taste exactly as it should taste. Control is everything.
The principal reason to brine is to provide extra moisture and prevent delicate flesh from drying during long cooks; brining allows you a little extra leeway in terms of cooking slightly more done than an ideal level of juiciness would otherwise allow.. Brining is also a somewhat tricky and finicky method of marinating to introduce flavors into flesh; injecting is generally easier and more effective -- but "generally" means there are some significant exceptions where brining works better -- spare ribs, for instance are such an enormous pain to inject, why bother?
Because goose has so much fat, brining is probably unnecessary to keep it from drying out, and not as effective in that sense as putting some citrus in the cavity, trussing, and rotating the bird during cooking in order to have it "self baste."
That doesn't mean you can't or even shouldn't brine goose, it just means you have to understand how to move flavors into the meat without moving too much salt or water. Boil the aromatics, add salt and sugar to make a mild to medium brine (i.e., not too much salt or sugar), add lots of spices while the water is still hot, short soak, lots of citrus.
Worth mentioning that brining is a very effective method of defrosting frozen duck and goose (and they most often come frozen) -- you might even say that defrosting in brine is killing two birds with one stone. Also worth mentioning that brined duck and goose seem to render their fat better, and consequently are usually less greasy after ordinary roasting (or smoking).
I successfully brine goose and duck, so you certainly can.
There are some people who don't care for brined poultry or pork, no matter how skillfully it's done.