But I still get dish soap, shampoo, conditioner, body wash, paper towels, and a few other items off Amazon since they have a better price.

If they weren't that successful then are they the best people to be dishing out paid advice?

Heads up, South Korea is now forcing Airbnb to offer full refunds to South Korean guests all the way up to the final day of the reservation. 

The dish washer sounded like it was at or near the end of its life anyway.

Then in the early 2000s the Koreans were complaining about the illegals dropping prices .

We leave a starter set of essentials like laundry pods, toilet paper, paper towels, trash bags, dish pods and soap, etc and then we rotate out local goods. 

Heck, I hand wash my dishes mostly at home anyway.I am targeting a February move-in.

I make sure they have enough TP, DIsh pods, Laundry pods, and Trash Bags for the entire stay. 

Should I make an offer with the qualification that all furnishings, dishes, pots, and pans stay, except for "personal" effects?

By all means give them a note on how to use all of the appliances (eg clean out the lint trap on the drier after each load, do not use dish soap in a dishwasher, change the furnace filter regularly, etc).