Hi @nerd , thank you for pointing it out, and I am sorry for making this mistake. I got carried away because I spend SOOOO much time exactly with this power management of the Pi. Also thank you for not “hanging” me from the tree
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The technical context:
Connecting to the GPIOs bypasses the power management circuit of the Pi. What is this? It is not much in terms of components (up to 4), but when it is missing it causes all the problems described. Pi 3, 4, and 5 have different levels of implementations of such a power management circuit.
In general, to be complete, it needs:
A) 2 x 5.1k resistors for the USB PD(Power Delivery) handshake. This is for two reasons. 1) To tell a USB PD power supply “Hey there, I need only 5V 3A, not 20V 5A”).
///edit: this is wrong: And 2) These resistors make the connection “hot”. So for a linear power supply, the capacitors are then charged up, not starting from “cold”. Meaning no issues with the Voltage ramp up/rise-time.///.
///Update: I am curious if a problematic linear power supply would still work in the Leeloo circuit, for other reasons now unknown. Got to get my hands on one.///
B) A switch (a P-Channel MOSFET) that has the function to completely cut power to the Pi, not leaving it in standby.
/// this is wrong: Because from there the Pi 3 and 4 cannot turn on again.///
///this is correct: They can, with the GPIO03 or Global_EN to ground shorting, and the Pi 5 with its switch and PMIC///.
///Why I wrote it this way, because I forgot to write what I this was leading up to: If you make an implementation using the “Short GPIO03 to ground” function to restart, then you would need two buttons. One for turning on, one for initiating the shutdown. Otherwise it would restart when you want to shutdown.///
///update: Moreover: In standby/halt the 5V PINs are still on in all Pi generations, and this is not what you want if a HAT is attached to it and a FAN that is supposed to stop spinning when the device is turned off.///
C) One TVS Diode for ESD protection.
D) A Fuse for overcurrent protection.
A) and B) are needed for correct function. C) and D) is for safety. Pi 3, 4, and 5 do not have B). This is the reason why when doing software shut down all Pi generations “end up” in standby, with the red light on.
Note: Pi 3, 4, and 5 have different implementations and also each generation has different revisions of what functionality the power management circuit has, eg. here is an interesting read about what happened on the first Pi 4s.
The Leeloo is not only a case, but it is a case with PCB that has such a power management circuit implemented. All functions, ABCD. Therefore, connecting a power supply of any type, /// update: this is a wishful claim: just works./// To be confirmed.
It is possible to hand-build/solder such a power management circuit, with wires and THT components. It is fun to do. If you prefer not to do it, the Leeloo was made for this.
Thank you, and happy building, and any questions about Pi and Power management, happy to respond. It is a rabbit hole so deep I never expected it to be 
Timo