OK, making progress... sort of.
Last time I probably made something wrong with the wiring, because when now I connected everything again (wires from the separate PS2 socket to the Leonardo, plug of the throttle quadrant to the socket), uploaded the compiled unmodified code to the Leonardo, and replugged it to USB, my laptop started to pick it up. Not without weird things, though... :-(
Weird thing #1:
Sometimes (once or twice out of every 10 attempt, or so) Windows says Unrecognized USB device. Another replug fixes it, so albeit annoying, but this is something one can live with.
Weird thing #2:
When not unrecognized, the Leonardo gets properly detected as USB Input device in Device Manager (sometimes it takes 10-20 seconds after plugging it in - maybe I did not wait this long when fiddling with it last time), and if the Throttle quadrant drivers from Logitech are downloaded, extracted and installed manually, it even gets detected as Flight Throttle Quadrant (USB). Hooray! BUT...!!!
After an additional 15-20 or so seconds, a yellow exclamation mark appears on the device name, and on the General tab of the device, the message "This device cannot start. (Code 10) The I/O request was canceled." appears. Also, on the Events tab, the message "Device USB\VID_06A3&PID_0C2D\0 was not migrated due to partial or ambiguous match." appears, with "Status: 0xC0000719". All this sound like if the throttle quadrant/Leonardo is not willing to communicate through USB.
My conclusion is that the potential cause is one of the following:
1. The Leonardo code has some sort of issue/bug. I don't think this is the case, as others have it up and running.
2. My particular Leonardo board has an issue. As it can run other code, and can be detected/flashed without issues, this might not be the cause either.
3. Something is still misconnected/miswired. I also think this is not likely, I have the green LED on the throttle lit, and USB connection is up, so it gets detected by the Leonardo.
4. USB port is not capable of supplying enough current. Not sure about this.
5. Something software on my laptop. Trying to figure this out. Win10 is up to date (20H2), and AFAIK in good shape, no issues with it.
Any help is really appreciated! I will drop a line if I get any further on my own.