So, it actually works like this with the LEDs: The LED that flickers when you send data from the viea serial monitor to your board is the RX LED, because the board RECEIVES data. And the LED that flickers when the processor, i.e. the ATmega644, sends (TRANSMITT) data to the serial monitor, is the TX LED.
However, I have found that the TX-LED, i.e. the LED that lights up / flickers when the ATmega644 is sending data, really only lights up, not during the reset, opening the serial monitor, or when connecting the board to USB. And since there is currently no sketch on your processor that sends data to the serial monitor, this LED is also not active.
However, it should by no means be the case that one of the two LEDs is active when the board is connected to the USB, only with the bootloader and without data from the serial monitor being sent to the board. Otherwise you either have a problem with the operating voltage or a mass problem. because in the idle state, RX and TX are at HIGH potential, so that the LEDs are off. I've used the CH340G several times. And also the ATMega 644 and the 1284 in connection with the CH340G, and that usually works without any problems. Maybe really a problem with the structure on your prototype board.