By "nowhere", I meant that the code was enabled, but there was nothing hanging off the Tx pin.
The only time that will get consumed is the write operation pushing characters into the Transmit buffer and waiting for the UDRE flag to get set (which will always be set). This still takes some time, but it'll be on the order of a few microseconds per character
And when you send another character when there's one already being sent, like in a multi-character debug string?
Every time you go to write a new character, you have to wait until the last one has been clocked out, so every character (apart from the first, assuming it is more than a character period since the last string was sent) will consume, at 9600 baud, over 1 millisecond, whether or not there is something at the far end of the serial line/USB serial/taut string to receive it.
void HardwareSerial::write(uint8_t c)
while (!((*_ucsra) & (1 << _udre)))
*_udr = c;