Hi,
Why the weight, and not the width, or the length, or the height ? or the meta data, or custom fields, or other fields of variants ?
I understand that it is useful for you, but we're limited in space for the variants listing and not everyone want that field there and most people don't care about the weight of variants as they have the same weight.
That's why it's not there. If you need it, the best will be to customize the view files displaying the listing of variants and add it yourself. A developer should be able to do that for cheap as a small custom job if you're not a developer yourself.