I'm quite confused. I visited again last week and there was no beverage tab on the menu page. I asked a server, and he said to order at the bar was the only option.

They use Joe coffee. https://joecoffeecompany.com/ I agree it's quite good! I remember trying one of the retail locations several years ago and didn't think much of it, but I've found it consistently great in the Sapphire lounges. LGA uses the same coffee, and it tastes great there too.