Howards Restore-a-Finish has a solvent that softens the original finish. It also has a varnish component and some types of it contain a stain. Just so you know what it does.
FWIW, I recently restored a pair of Grundig speakers, circa 1960. The light walnut veneer had the typical water marks, scratches, and flower pot stains. One speaker had the veneer baked (as sitting in the Sun can do for 20 yrs). I used 800 or 1000 grit wet-and-dry sandpaper and polymerized tung oil to recover the finish. The final coats were applied with a rag. Even the BIL, a professional furniture builder and upholsterer was impressed.
About steel wool: I've probably used a few hundred pounds of it over the years. With the advent of 3M scrubbing/sanding pads, steel wool is obsolete. There's no reason to mar your finish work with particles of metal when a synthetic pad works as well and leaves no metal bits.
Mayonnaise and ashes? Hey, it's not 1930 again. Use products made for refinishing wood.