DIY Furnace cannot reach set temperature
If the contactor does not drop out then the problem is entirely with the heater/oven, assuming your temperature readings are accurate. All the controller can do is run the heater at 100% duty cycle.
At 1325°F (~700°C) you can estimate the temperature visually, it would be dull red roughly. That should confirm the thermocouple reading.
Nichrome increases in resistance with temperature, and heat loss also increases so you may  have to increase the wattage and/or decrease the heat loss.
Your discrepancy is relatively substantial. I did not watch the video, but if you are emulating what was done fairly closely, I wonder if you are using 120VAC and the YT gent is using 240VAC. That would be 4:1 difference in watts and could account for the difference in performance.
Edit: Briefly looked at it and it seems like the YT video used a similar length (he says) of AWG 22 wire and is also operating from 120V according to the comments. It does not appear to be an electronic problem. There is nothing wrong with your circuit diagram.
Please confirm whether the contactor is remaining on.
Note that according to @mkeith the video shows 1 hour to get to 1100°C, so it would not be able to ramp as fast as you have set and the controller may shut down as a result (could be a safety feature or simply running out of program time). This is a result of insufficient control authority (in this case, heater watts). In some kiln controllers I designed we would shut down if the loop stayed out of control for more than a certain time, which minimized the damage in the case of a shorted thermocouple.