"...having a concrete driveway would totally destroy the lovely look and feel of this pseudo-statepark trail as our driveway." I disagree with this premise on two counts. As I view the photo (taken midway along the drive) I think some of it looks lovely. The foreground clearing looks nice. The forest and the distant opening looks nice. But it looks uninviting for passage by any vehicle other than a farm tractor. I would fear for my car's paint driving through those weeds and close brush. It looks only inviting for a pedestrian who is wearing jeans, not shorts. To further your example of a state park... all big parks use asphalt (sometimes concrete) roads and, in my opinion, their look is not in the least offensive. (Those solutions are driven by cost and practicality) In fact, it's not the presence of objects right in your face, but the CLEARING AWAY of nearby objects (that the road and it's cultivated--usually mowed--edge creates) and the removal of brush and low-hanging limbs, that makes a space look inviting for the passage of people and vehicles. Personally, I think the choice of material of the drive isn't as critical as the clearing of objects and the creation of space. If your drive was of aggregate it could be pretty. If it was asphalt or concrete (including "stamped" and/or colored) it could be pretty. If it was of brick, it could be extra pretty. What man creates for himself will not look like what beavers or raccoons create. Their needs are different. Somewhere along the line, practicality will play a part so the solution needs to work as well as it looks. My illustration is not to show or suggest a material. It's only to show that creating a little space alongside and above a passage way makes it look more inviting for passage.

You need a weed torch. Good top kill, though perennial weeds will have to be burned off as they return. Heck of a lot faster and more convenient than boiling water. I don't know of any non chemical method of weed control that will translocate down through the roots. Here is a link that might be useful:

Do you know if they are annual or perennial weeds? If they are annual you could try throwing out some pre-emergent. Maybe its not the same weeds coming back but new seedlings. I like the blowtorch idea. Sounds like fun!

If you want to save your soil, put them in a container of water. After a bit, swish it around. Some of the dirt will fall off, and you can keep going till you're happy with the amount of dirt you got. Throw away the plant and roots, and use the water for some of your plants. The dirt/mud will fall right back into your yard.