I have 2 sweet autumn clematis vines in my front yard: one grows on netting into a walnut tree, and one on a net around a utility post.
They have gotten so weedy and unmanageable, even though I cut them to the ground every year. How can I kill and remove them? The bottom of these vines are downright thick pieces of wood. They drop seeds everywhere and even the seedlings are very vigorous in my garden!

Sounds like you're going to have to dig them up. I wish I had that problem - I love clematis.

I'd try posting an ad on your local freecycle or craigslist site. "Free clematis - U dig". I bet someone would be glad to take them off your hands.

I know how you feel though... I have a clematis that I swear, during the summer, I am cutting pieces off of at least weekly so it doesn't strangle nearby plants/trees! It's definitely some aggressive stuff. Good thing it's pretty!

Unfortunately, as lovely & sweet-smelling as "Autumn Clematis" is, it's now considered an invasive in most, if not all, parts of the U.S. Thus, if I were you, I'd probably just try to eradicate what you have rather than try to rehome it.

I was thinking for a long time of planting some, but am rethinking that after seeing a neighbor's few plants start popping up in other areas of the neighborhood where it hasn't been wanted.