Not all scooters have the over steer you're referring to. I find alot of the late 80's and 90's scoots with telescopic forks tend to have a steep steer tube angle (low rake numbers) combine with very light front ends. This gives the oversteer characteristics you're talking about.
The larger scoots, especially anything with a trailing link suspension and longer wheelbase feel great. I was worried when I bought my Lambretta Cento (because it's VERY front lite) that it wsa going to be awful to ride, but the bent/raked fork assy with the trailing link suspension makes it way more stable than most mopeds I own with 17 in. wheels.