The old (maybe incorrect) advice I was always told was that continuing to heat the coffee fouled the taste, hence the recommendation to use an insulated mug/thermos as opposed to a Mr. Coffee style warmed carafe. What makes this gadget different/better?
That makes sense. The mug version of Ember (https://ember.com/products/ember-mug-2) is open on top, which seems like a fancy Bluetooth version of an old-school mug heater.
I could see the sealed one (the "Cup") as working better though.
Much of coffee's flavour comes from a balance of acids and oils. These, particularly the oils, are sensitive to temperature. But unless you're drinking your coffee all day, you're unlikely to notice it. Real spoiling sets in after several hours at the kinds of heat you're probably running your Ember mug at. Most drip brewers with heating elements for the pot keep the temperature too high (often just below boiling). A steady heat is much better for the flavour than reheating. (This is what I remember from working at a coffee shop way, WAY back when I was at university.)
It would foul the taste because in an open carafe it distills the brew via evaporation.
If you don't let the water escape the taste stays fairly stable.