A comment by Andy reminded me of the fact that a standard Arduino (non SMD types) comes with the FTDI chip and therefore can be used as a USB to TTL serial programmer too :-)
To make it work you must carefully remove the AVR chip. Take a small and flat screwdriver and gently insert it at the left/right sides of the socket and lift the chip step by step. Alternate back and forth from left to right until the chip comes free. Make sure not to apply too much force, as you WILL bend the pins if the chip comes free unexpectedly.
It may also work to connect the Arduino’s reset pin to GND (1k resistor) to make the onboard AVR chip go silent and not respond during the programming, but I haven’t tested that.
Now program the board as described here.