I was able to test the Arduino Nano by flashing a sketch with avrdrude and the USBtiny and it works but not very practical.
Keep in mind that when you do that the bootloader is erased so you always need to do a Tools > Burn Bootloader after doing that in order to go back to being able to do uploads over serial.
Domos:
The solutions seen on this forum talk about installing the driver of the ch341 but, this one is already part of the recent Linux kernel.
If the port of your Nano is appearing I think that indicates the driver is working correctly.
Budvar10: @pert
He wrote that he put the bootlader back.
I don't see where it says that but I do think it's likely Domos already knows this, still worth noting just in case because I've seen that trip people up quite a few times.
Other common issue is using charging USB cables or maybe just a bad cable, which I'm guessing would be indicated by the loopback test results.
kprims:
I change the bootloader to Arduino Uno on my Nano's. Might be worth trying on yours. I do the same for the 16 Mhz Pro Mini's also.
I have test with your solution but same problem.
Not very optimistic, I just checked that my old Arduino Nano that works also uses the ch341-uart driver so the problem does not come from there.
I have flashed a sketch with the ISP programmer which tests the serial port in reading / writing, it works but the reception does not do as expected, I receive the characters by "batches" as if an intermediate buffer was emptied.
The same sketch on the functional Arduino Nano displays me character by character for each "println".