If you tried to only refinish the chequered areas, I don't think that you'd get a match with the rest of the stock - so it wouldn't look right. Also, if you use stripper (and a toothbrush) on the chequered areas - it would be difficult to keep the stripper from the other portions of the stock.
I think that you'd be better off in stripping the whole stock and refinishing it. Using the strippers available today, it's not that difficult. Just follow the directions EXACTLY. (A toothbrush IS good for getting the stripper to work in the chequered areas.) As to a new finish - you've got a world of things from which to choose..... an oiled finish.... stain & polyurethane.... paint.... stain & varnish, etc. If it were me, I'd choose stain, plus a satin polyurethane... which will give the best protection against future "wear".