The reason steel wool is nice is it tends to scrape off the gunk whereas pressure with a cloth may tend to press some down into the grain.I didn't use steel wool, instead going with the instructions on the web site of using a soft cloth. I had to wipe it down fairly hard first (without oil) to get rid of accumulated gunk as this bass gets played a lot. Then wipe on, let sit for a few minutes, wipe off. I went across the grain but since I was using cloth instead of the steel wool I didn't notice any issues.
There's a few ways I prefer to deal with gunk on an unfinished board...
1.) Don't let it build up in the first place, by cleaning the board every so often with a safe cleaner oil.
2.) Clean it off with light pressure steel wool with/without cleaner oil. Just watch out for steel fibers around pickups.
3.) Clean it off with light pressure cloth with cleaner oil.
In all three cases, however, do NOT let the cleaner oil sit on the board. Always wipe it as clean/dry as possible as soon as the gunk removal is complete.
And don't forget to also use an actual board conditioning oil as needed as well. All open grain boards will need oil at some point, and even being quick to avoid letting cleaner oils sit on the board, they will tend to accelerate the need for re-oiling.
IMHO/IME!

I should get the bass in 3 weeks (too much time jejee). I'm not really concerned about it's value in the future, I really hope to keep it for many years and kill my GAS