Ducks on wire is fully doable, according to Dave Holderread (#1 duck expert, who also adds that he thinks it makes life less enjoyable for them). Many people raise call ducks this way, but it can be done with larger ducks too.
Personally, besides being less enjoyable for them, I think it makes the meat & eggs less nutritious & delicious if the birds are not allowed access to green foods and insects that they would find if given pasture.
However, I think there are efficient ways to reduce the mess & still have reasonably happy & healthy birds producing good quantities of meat and eggs. Holderread places water containers on wire "water porches" at the opposite end of the runs from the food. The birds will mostly hang around the water, up on the wire (with significant drainage below), and leave most of their mess there. But they get exercise & forage opportunities going back and forth between the food and water, and they still have access to earth & soil & bugs.
As for water, you are correct that they need to dunk their heads to clean nose & eyes (chicken waterers are not adequate). They do not *need* bathing water, but they are happier and cleaner if they have it. Swimming water is definitely not strictly necessary, and ducks raised without swimming water generally won't show an interest in swimming even when given an opportunity. That's not to say that swimming is not beneficial or good for quality of life, but it's definitely not a strict necessity for a decent ducky life.
If you decide to raise them on wire, make sure to check the specs recommended by others who do it (Holderread's books have specs) in terms of size of wire openings. Too small, and poop won't flow through effectively. Too large, and it will injure their feet and lead to health problems.
Good luck with whatever you decide. This is the best place to get info, plus Dave Holderread's book: Storey's Guide to Raising Ducks (now in a new edition).