I follow much the same procedure mentioned, however, I skip using steel wool until there is some finish on the wood.
I have a 22 with a mahogany stock, so it has lots of open grains that get the little bits of steel wool in them. The tighter grain woods should not be an issue, but it's something to think about is all.
I like the handrubbed finish myself such as tung oil due to it being very easy to make a great finish, but it takes patience.
I refinished a stock for a friend that wanted a glossy, water tight finish on it. I live in the PNW, spelled, rainy hunting seasons. I used a spray can Varathane gloss. I put one or two thin, thin coats in the bedding area to seal it well, taking care not to add any dimension to the bed as best I could.
I then protected this area with some masking tape and proceeded to finish the stock with 3-4 coats of finish, using steel wool in between coats. They turned out very well and my friend was absolutely elated at the results. He was so happy he had another stock in my hands that week to refinish for a relative of his for christmas. Again, got rave reviews from the customer.
The one trick I found to clean out the checkering without removing too much of it, if any, was to use a utility knife razor blade to scrape out the finish in the checkering shortly after the finish remover was used. It was slow and painstaking, but it provided superior results. On that note, if you want to refinish a checkered stock, you may not have many staining options because it is pretty difficult, if not impossible, to get the stain color out of these checkers because they soak up some much stain in original finishing process.
I have quite a woodworking background and refinishing stocks is probably one of the most rewarding projects I've ever done because you get to see a beat up old stock turn back into a total gem, quite satisfying and worth the time and effort to do it right. Which leads me to an old shop saying to operate by. "If you don't have time to do it right the first time, when will you have time to fix it?"
jeepmor