Have you checked to see whether there is a pan? (i.e. dug down a few test sites) There may be no pan, and it’s just a clay layer or so.
Secondly – when does it get waterlogged? If it’s during winter, then I suspect willows won’t be of huge help then (although planting any trees there will improve the soil) as they’ll have gone dormant for the year and won’t take up any water. If they have a plentiful supply of water in Spring – they have no incentive to send out deep taproots and will just send out lateral roots (willows are famous for lateral roots).
Depending on your local climate, evergreen trees may have a longer growing season than deciduous and may still be uptaking water during winter.
I suspect I may have similar conditions on a site that I’ve yet to plant (in New Zealand), and I plan to experiment with Gunnera (if possible, I have to check regulations again – for example G. tinctoria is banned but I think G. manicata isn’t) for this case. Gunnera have been documented to grow for 11-12 months during the year nearby my site (which gets close to zero, but rarely frosts), and are extremely vigorous (they fix nitrogen using Gnostoc bacteria – a little-known extra class of N-fixers that most people don’t use) and are edible too (wikipedia claims they are sold on the street in Chile). If they can grow during winter and uptake water, then hopefully they can be chopped to the ground during Spring and Summer to leave the water available for other plants (i.e. in effect – ‘turning off the pump’).