Those aren't hard to strip, browning are a pita. I use Circa 1850 as Kamlooky said, A small can goes a long way; use a plastic putty knife to lift the slop off, get some 0000 steel wool, for final wipe down, burlap will work also, get some blue nitrite? gloves and stiff tooth brushes for the checkering. don't rush it, varnish comes off fast, some of the modern stuff takes a while and few coats.
And as you say, don't sand, or you will have to do the whole thing, and any little marks will show up, and show in your finish