Maybe I'have solved this issue.
But wait before clapping hands, I have to check if the sketch works properly once soldered on my PCB with all the rest of the circuitry around.
The key seems very simple: you have to completely remove the Pro mini from your breadboard or PCB before attempting to do anything with it.
It doesn't matter if you want to burn the bootloader or simply update your sketch with a new version. Get it out from your circuit in any case. Then connect the wires from the USBasp dongle directly to the opportune pins. Nothing else should be attached/connected to the Pro mini when using the programmer. Simple as that.
I made a lot of power-cycles tests: the arduino restarts in any condition of voltage status. Zero issues at all.
Also, I allowed the discharging of all of the inner capacitances during the night. When I powered again this morning, it has started flawlessly.
I also repeatedly uploaded the same sketch (32726 bytes) six times using the above technique. Again, zero issues.
As stated in the beginning, now I have to test it into the "real world" before considering this problem closed. But we are in the weekend and my family needs me as driver! 