The thermocouple you linked to does not have an exposed junction. The probability is very high that it has an insulated/ungrounded junction, as grounded junctions seem to be much less common. If you have a multimeter, you can check whether or not there is continuity between the sheath and the wires to the junction. No continuity means insulated/ungrounded.
The elements were being switched by the controller in response to the control thermocouple junction temperature, which was insulated from the thermocouple sheath by around 1/16" of MgO powder and therefore the junction temperature lagged behind the sheath temperature.
What the logger traces seemed to show (and I completely failed to explain),was that the thinner thermocouples followed their sheath/surface temperatures much more closely, having either much less insulation or none at all.
The control thermocouple was at the top, in the middle of the oven. A blade would have been in the middle of the oven with its edge either up or down and the whole area of the blade exposed to the radiated heat of the elements.
Rather like the hood of a car on a sunny day, the large area of steel (probably black from the quench) will reach a temperature significantly higher than its surroundings as a result of the radiated heat. The different thermocouple diameters were my attempt to simulate a number of different thicknesses at different points on the bevel.
Once the control thermocouple junction temperature approached the setpoint, the PID terms kicked in and started backing off the heating. The thin thermocouples (or the blade they were intended to simulate) then started to cool as the amount of "off" time in the 2-second output cycle increased, until everything in the oven was at about the same temperature.
The 6mm/1/4" thermocouple that I used in my first oven was just one I had to hand before starting the build. The testing showed it to be unsuitable and I subsequently bought something more appropriate for the task.
I actually bought a Mineral Insulated thermocouple with a grounded junction (the junction is welded to the inside of the sheath) and further testing showed the response to be adequate for my purposes (YMMV). Unless you are in a position to do your own testing, I would strongly recommend an exposed junction thermocouple. It wants to be somewhere in direct line of sight to the elements.
The idea is that the control thermocouple is the fastest-responding thing in the oven and will back off the heat input before the blade(s) overheat due to the radiated heat from the elements.
My second trace was effectively an attempt to slow everything down, using a ramp, until the control thermocouple responded just about as fast as everything else in the oven. Once this was the case, the other thermocouples (simulated blade) did not suffer nearly as much overheating.
If you change your thermocouple for one with an exposed (or grounded) junction, run it up to maximum working temperature before you run the Autotune. This puts a nice dark Oxide coating on the thermocouple, making it more responsive to radiated heat than a shiny new one.
Another couple of questions: What is the element power, how long is the output cycle of the controller and what was the controller output when you measured?
The reason for asking is that the radiated heat is only present when the elements are powered. If you have low powered elements, they may be on for a large proportion of the time.
Up to a point, shorter output cycle times are better. During my testing, I found 2 seconds about the best setting. It gave measurably better stability than 5 seconds. I also tried one second, but could not detect any improvement, so went back to 2 seconds.