Laika, when I visited the Netherlands I saw people carrying their friends on those racks. It is hard to believe that a milk crate of groceries would be more difficult to carry than a person sitting side saddle.
The bike pictured in the thread "Panniers without panniers" actually feels more stable with a heavy load but it is a touring bike. I also have a "Dutch bike" but made by Trek and have owned a German made city bike very similar to the dutch bikes. I don't notice the effects you describe. Is the rack loose? Tires inflated? Milk crate attached securely? Does the load shift?
For the tinkerers, the German city bike I had, had a spring running from the bolt where the front rim brake attaches going back to the down tube. I'm out of town right now so can't get a photo of it. When you turned the handle bars to an extreme angle the spring would extend to make it more difficult to steer further. This was handy for those times when you have it on a kick stand with a heavy load and don't want the front wheel to turn too far. I'm not sure if it had a positive effect on handling but I have a broken clavicle to show what can happen when the wheel turns too far while rough riding.