Yeah, I share your annoyance. Given that this thread is unlikely to be deleted, let's try turning it into a helpful one, instead.
Just a forewarning, I'm not really sure what the solution is, but I'd like to try to help.
Have you tried increasing the "minimum force" setting? I have a T300 (belt-driven wheel) and it doesn't suffer from shaking while idling. However, it did shake wildly when I removed my hands while driving down a straight. I found that increasing the minimum force (from 1.0% to 6.0%) eliminated that phenomenon, allowing me to remove my hands from my wheel.
And as an aside, I increased the minimum force in rFactor2 (I know, different game...) from 0.0% to 4.0% and it eliminated the steering wheel vibration felt during idle.
Since I'm not really an expert on this, I'd like for you to help me confirm whether "minimum force" is even the correct setting to be mucking about. Start off by increasing it to a wildly high value, like 50% or so. If it does eliminate your issue, then we'll know that it's the correct setting to change. From there, it's all trial and error in determining a proper value that will work for your wheel. It's likely to be below 10%, but I might be wrong; I've never had a Logitech wheel.