So, after ~2 hours of elbow-grease, sweat and plenty of Jif the "old" Weber is almost good as new. It is amazing just how good it has come up. The only thing that remains is a couple of coats of high-heat "shimmering rose" paint, and some varnish on the wooden handles.
We put a small roast on it for a test run, and it worked beautifully. Great stuff!