The output that you showed is for the old bootloader (baud rate 57600). There are two types of boards with the old bootloader; the older original Nanos with FTDI chip and about all clones with CH340. What is the chip pn the bottom of the board?
If it's the FTDI chip you can run the loopback test (Loop-Back Test Instructions); for the CH340 that test is non-conclusive.
If you press and release the reset button, do you get one quick flash (indicates old bootloader) or a couple of quick flashes (indicates new bootloader). Or no flashes at all in which case the 328P is malfunctioning.