Refinishing oak table and chairs

Well this is my first project involving furniture refinishing. I decided to take on the project of refinishing my oak kitchen table and chairs. It is somewhere between 15-20 years old, so it's definitely had its wear. The plan is to restain the table top and chair tops using minwax golden pecan, and then sand and paint the table legs, chair legs and back a satin black.

I've been using Klean-Strip stain stripper to remove the old finish, but it seems like I'm not getting the progress that I should. Right now the middle leaf in the picture below has had 3 applications of this stain stripper. I've used a plastic putty knife to scrape off the application of stripper for the first two, and then for the third I used some fine steel wool.

I'm wondering if it would be better to just go the sanding route? Or just keep my patience and keep applying the stain stripper until I'm able to re-stain?

Thanks.