Nate:
Hats off to you, because you have definitely solved the mystery: "What brand is this?" Looking at your photo above, which appears to be a later model than the unit chez Draper, the indicia, their fonts, everything is a match.
However, we are still left with other mysteries, which may not be solved anytime soon. The site you linked to is quite interesting, but the author makes a very simple, yet critical mistake when they assume the dishwasher is at least as old as the acquisition of Gaffers & Sattler by Magic Chef.
The Website that author links to in support of their theory that the DW goes back to 1969 at the latest actually says that Magic Chef acquired G & S in 1969, not that there were no more G & S-branded appliances after that date. In fact, both the Draper DW and the one in the photo you posted look very much like "acquisition specials' - cheaply-made appliances using a once-exalted name. 1950s and early 1960s Gaffers & Sattlers appliances I've seen pictured are obviously well-made, expensive, luxurious appliances. These two dishwashers are no such thing. And I know enough about their D & M DNA to suspect that they're later than 1969.
I am under a distinct impression that Gaffers & Sattler was very much a West Coast brand, not often seen in the East. I have never seen one of their appliances in person. If that is true, that would put Betty Draper's dishwasher rather a long way from home.
At any rate, we don't know everything yet, but we have an excellent start, thanks to you, Nate. Cheers!