No your geese won't sink - you could have swans & they wouldn't as all waterfowl have oily downs & feathers & skin & if you've ever mixed oil & water for example ship oil spills or washing oil on a frying pan in the washing up the oil & water separates with the oil on the top - because it's boyant - this is essentially why waterfowl float whearas if a chicken falls in the water trough it's game over if no-one is around as the water saturates their coats causing it to become heavy but this doesn't happen with waterfowl. Also if you ever rear any for meat feel the skin & look at the amount of fat that comes off the bodies of geese & ducks that's why you don't need much to cook them with.
Ours are very lucky as we have a river going though our land & they won't go on it when it floods but in general they go for a couple of swims & spend a lot of time grazing often near the ducks or they will go off up the river for most of the day - it just depends on how the mood takes them. I find though they do spend less time on the water than the ducks, but I've seen them diving under often - the youngsters can be a bit nervous of it at 1st before going on the river but they love trays of water as goslings. Every morning though the geese get excited when let out in the morning &the 1st thing they do is run & do what little flight take off they can at the edge of the rivers bank into it honking loudly for a bathe, & hate it if left in unable to bathe for a few days.
As for mating with ours I wouldn't know I've only seen it once in 8yrs & I think it was on water but ours seem to always be very secretive about mating as do our turkeys unlike the hens & ducks
but we nearly always have great hatch rates with them.
If I were you I've give them a nice big area around 1-3ft deep which is easy to get out of like the pond your suggesting & they'll love it x
P.S Harry I don't really associate with your following quote:
" Mind you, they do that when you feed them, when you break the ice for them on their water, struggle up a hill with 25l container of water for them.......basically any time you do anything for them they fix you with a beady eye and say 'I really really hate you"
It's just ours seem to be pretty ok &polite but have an answer for everything, but the gander doesn't like you getting too close i.e a few feet (Dad got s love bite once trying to catch him & everyone down the pub thought it was my mum
) & you occasionally get pecked by the females if trying to get close when broody but thats it really & we get a nice chatter every morning & when we leave food late at night.