Thermocouples drift for a lot of separate reasons. As the wires thin (the voltage is created by the temperature differentials along the wires, not by the tip) the amperage that they create lessens. They have to be very very thin before this in and of itself is going to cause much drift that matters.
The gizmo that reads the voltage, an analog to digital converter, generally has a very high input impedance. The cheap chip that I use in microcontroller projects has 60,000 ohms impedance. A 1Ω change is going to cause a very small drift in measured temperature. The resistance at the tip is going to have to change a lot before this is significant. I would only expect a problem right close to failure. I am unsure if kiln controllers monitor the resistance of the thermocouple or if they just detect open circuits. Its a good question for Bartlet Control or Skutt Technicians.
Much of the voltage output drift can happen because the alloys change over time when hot. This is called alloy drift. While it happens with type R &S thermocouples, especially if not in a specially designed protection tube, it is a much faster and more significant problem in type K thermocouples. I believe, but do not know, that it also happens in the lead wire if it gets too hot. Thermocouple lead wire should not contact the kiln case. I have wondered how much of the drift is caused by corrosion at the joining of the dissimilar metals at the thermocouple and the controller board.
Over the last 20 years I have come to the belief that all kilns should be equiped with Type R or S thermocouples. They are long lived, and much more accurate. They drift slower. They are much more expensive. But they are less expensive than a load of ware.
I had the joy of using an very expensive Nabertherm branded kiln for a number of years with a type R thermocouple. I used witness cones in each firing.I kept the last set on top of the kiln and compared them. Mostly the firings were cone 9. I adjusted the top temperature each firing. Often this was less than 3 degrees F. It was never has high as 15 degrees F except the first firing to a new temperature or a radically new firing schedule near the end point. I think at the very least you should be able to see the actual speed a kiln fired at. I suspect new better controllers on kilns have the ability to see and record this.
Other sources of drift are slower firing speed as elements get slower. The controllers on US kilns to my knowledge fire to and end temperature only, not a cone. So if you have a fast speed set and your kiln slows down you can end up taking more time to get to that temperature and then you are at a higher cone. Because of this I think that people should slow down the last few hundred degrees of their firing to a speed significantly slower than a new set of elements is capable of. The second part of this is that a cooling profile should be used that is slower than a full kiln cools naturally at. It does not need to be much slower. Older elements is frequently the reason the cones get more mature as the kiln is used more.
This is the most detailed description of my understanding of this issue I have written. There is more that needs to be said including kilns getting leakier, ice point adjustments, non-linearity in thermocouples, how to slow alloy drift. I know nothing about drift in Analog to Digital Converters. Likely they are less perfect over time than we acknowledge. Texas Instruments has a paper on this, https://e2e.ti.com/…/adc-accuracy-effect-of-temperature… I have not read this.