Nigel:
I don't buy your phase angle fired used for small loads. 100 A isn't small as described here:
https://www.eurotherm.com/products/power-control/power-controllers/7100a/
PID control, just basically makes the setpoint agree with the measured value and compensates for undershoot/overshoot. PID requires step changes in the setpoint to set the tuning constants. I have not done PID control with auto-tuning, but have implemented PID in software. I also did some wierd PID control by controlling the proportional band (setpoint) of another controller. i.e The controller was designed to control the oven temp, but I wanted surface temp, so i read the surface temp and used my own PID to modify the oventemp.
Auto-tuning controllers are great. In general, the PID controller might output a 4-20 mA or voltage signal and the load controller implements the phase angle firing, or slow cycle triac or Zero cross pulse. Phase angle firing gives you the finest control. Slow cycle triac gives you the lowest interference and slow cycle triac is essentially relay control.
The power controller can be just that. It can control power or it's input is usually proportional to V^2. Furthermore there may be "tungsten" controllers where the load is non-linear. In order for the power controller to operate into an inductive load, one has to incorporate current limit.
Finally, in one system I designed we used 1500 W DC power supplies for low voltage custom tantalum heaters. This allowed EASY computation of power.
Previously, we used phase angle fired controllers operating into variacs. The new design made the panels smaller and we didn't need to build custom hardware (Variac/meter panel) and the values were actually correct for actual power measurement. The panel space was 1 RU (Rrack Unit) for the power supply/metering and 1 DIN for the controller.