Sycamore trees have a very long life span. If yours are actually dying, there is nothing you can do for them. Try posting some pictures on the Tree Forum.

Brick pavers would be my first choice.

I would be happy with no kind of mulch (stone or bark) right outside my back door for use as a patio. Brick or concrete pavers would be a good permanent fix. A temporary fix could be some old carpet put face down or some outdoor carpet attached to some kind of a frame that it could be fixed to the earth. The latter is uncharted territory.

You could build a frame and make a wood deck. Build it right and it will support BBQ's, tables, chairs, etc. At the same time it will look great and you won't have a problem of weeds growing in it like you would with any kind of paver, pebble, woodchip, etc. You can even build the deck around or half round your trees and attach benches. That looks awesome! We have 9 decks here and our area is rustic California casual. This is our BBQ deck. It was the least expensive of all. It's made with pressure treated wood and unfinished. It holds two BBQ's. We could easily put a table and chairs, but we have those on other decks.