I've measured the heater resistance and noticed that it's smaller than the resistance for the rated Voltage/Wattage.
The measured value is around 4Ohms. As the soldering iron is rated at 24V:
P = U² / R
P = 24² / 4 = 144W
There it is. The soldering iron is rated at 24V 50W, but if you just connect a 24V supply, it will probably kill itself. You need to control the power delivered to the iron.
There's no need, then, to use a 24V power supply. The unmodified 19V notebook power supply should do the trick.
I've measured the resistance of the heater while hot, to how much it changes with temperature, but it raised to 7Ohms only at 450C.
That said, I've moved the power supply project logs to a new project, and will move a little faster on this project, as I have some other projects on my backlog.