I would suppose that you wood welcome japanese knotweed into your garden then??? Do not get me wrong I know a sycamore and have nothing personal against it!!!
The definition is there for a purpose and native would mean that the "native plants" have evolved togeather over time to co-exist. The definition of a weed is simply a plant in the wrong place, and in a managed landscape sycamore can be as such.
I am lucky enough to have a semi natural/ancient woodland which I am in the process of restoring to a ash/hazel woodland, native if you will. I spent many years studying the area before I decided what to do. There are mature sycamore of the woodland edge and I will not be touching them (I have learned to live with them). As you will know sycamore is a prolific seeder, (my wood floor is littered with them at the moment) seedlings open up large leaves earlier blocking the light for the native flora and thus leads to their demise, whereas ash opens much later leading to a happy native union. In addition sycamore seedlings and trees prevent the germination of ash seeds. I am removing the sycamore within the woodland but allowing them to regrow as coppice, hopefully they will not set seed in that time (remains to be seen). I felled 6 or so tall thin mature sycamore about 4 years ago and the number of native local provenance ash that came up in the clearing was gratifying.
1000 years is not that long in the life of trees, take an oak for example, 300 years to grow, 300 years to mature and 300 years to die. It is well known that it takes at least 3 generations to breed out inherrant xenophobic racism, so only another 2000 years to go.